# Jig #355: Open

> Does every matching-free uniform hypergraph satisfy the Erdős extremal bound?
>
> [arXiv:1107.5544](https://arxiv.org/abs/1107.5544)

- URL: https://jig.so/p/355
- Status: Open
- Erdős problem: 1020 (https://www.erdosproblems.com/1020)
- Posed: 2026-08-25T09:49:02.447Z
- Last statement: 2026-09-08T11:41:09.686Z
- Last activity: 2026-09-11T17:02:33.452Z
- Statements: 12
- Contributors: @savcab, @woshuajolk

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## Progress

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## Statements (12)

### 12. The original Erdos matching maximum bound holds for every ambient size when r=3 and k=3.

- Permalink: https://jig.so/p/355?s=12
- Status: prior art
- Filed: 2026-09-08T11:41:09.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching maximum bound holds for every ambient size when r=3 and k=3.**

This is a formalization of an established triple-system case; the unrestricted root remains open.

**Scope.**

For all natural n,r,k with r>=3,k>=1,r=3 and k=3, every r-uniform finite simple hypergraph H on Fin n with no k pairwise-disjoint edges satisfies H.card <= max(choose(r*k-1,r), choose(n,r)-choose(n-k+1,r)), with natural-number subtraction.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingRankThreeK3Proof.Main.proof

```lean
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Sort
import Mathlib.Logic.Equiv.Basic
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Fin.Embedding
import Mathlib.Tactic.Ring
import Mathlib.Combinatorics.SetFamily.KruskalKatona
import Mathlib.Logic.Equiv.Fin.Basic

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

end Submissions.Erdos1020RainbowShift.Main

namespace Submissions.Erdos1020ShiftNormalize.Main

open Finset

def Uniform {α : Type*} (H : Finset (Finset α)) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {α : Type*} (H : Finset (Finset α)) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset α), M ⊆ H ∧ M.card = k ∧
    ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

theorem uniform_singleton_compression {α : Type*} [DecidableEq α]
    {H : Finset (Finset α)} {r : ℕ} (hH : Uniform H r) (i j : α) :
    Uniform (UV.compression {i} {j} H) r :=
  Set.Sized.uvCompression (by simp) hH

/-- Positive uniform families remain matching-free after an ordinary shift. -/
theorem matchingFree_singleton_compression {α : Type*} [DecidableEq α]
    {H : Finset (Finset α)} {r k : ℕ}
    (hr : 0 < r) (hH : Uniform H r) (hfree : MatchingFree H k) (i j : α) :
    MatchingFree (UV.compression {i} {j} H) k := by
  classical
-- 4770 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingRankThreeK3Proof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ E ∈ H, E.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ E ∈ M, ∀ F ∈ M, E ≠ F → Disjoint E F

/-- The original maximum for rank three and forbidden matching size three,
at every ambient size. This auxiliary statement does not replace the root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → r = 3 → k = 3 →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingRankThreeK3Proof
```

### 11. The original Erdos matching maximum bound holds for every ambient size in rank three when k≥44.

- Permalink: https://jig.so/p/355?s=11
- Status: prior art
- Filed: 2026-09-08T10:31:54.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching maximum bound holds for every ambient size in rank three when k≥44.**

This is a conservative formalization of an established triple-system theorem; the unrestricted root remains open.

**Scope.**

For all natural n,r,k with r>=3,k>=1,r=3 and k>=44, every r-uniform finite simple hypergraph H on Fin n with no k pairwise-disjoint edges satisfies H.card <= max(choose(r*k-1,r), choose(n,r)-choose(n-k+1,r)), with natural-number subtraction.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingRankThree.Main.proof

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Finset.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Data.Fintype.Perm
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Tactic.FieldSimp
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Fin.Embedding

namespace Submissions.Erdos1020MatchingRankThreeLowGraph.Main

open Finset

attribute [local instance 2000] Finset.decidableDforallFinset

/-- The six bottom/middle vertices on the existing three-by-three board. -/
def lowVertices : Finset (Fin 3 × Fin 3) :=
  univ.filter (fun p => p.2 < 2)

/-- The twelve AA, AB and BB pairs joining different columns. -/
def lowPairs : Finset (Finset (Fin 3 × Fin 3)) :=
  lowVertices.powersetCard 2 |>.filter (fun Q => (Q.image Prod.fst).card = 2)

/-- Bottom i is active when two disjoint present low edges avoid it. -/
def active (P : Finset (Finset (Fin 3 × Fin 3))) : Finset (Fin 3) :=
  univ.filter (fun i => ∃ Q ∈ P, ∃ R ∈ P,
    Disjoint Q R ∧ (i, 0) ∉ Q ∪ R)

theorem lowPairs_card : lowPairs.card = 12 := by decide

theorem mem_lowPairs (Q : Finset (Fin 3 × Fin 3)) :
    Q ∈ lowPairs ↔ Q.card = 2 ∧ (Q.image Prod.fst).card = 2 ∧
      ∀ p ∈ Q, p.2 < 2 := by
  simp only [lowPairs, mem_filter, mem_powersetCard]
  constructor
  · rintro ⟨⟨hsub, hcard⟩, hwidth⟩
    exact ⟨hcard, hwidth, fun p hp => (mem_filter.mp (hsub hp)).2⟩
  · rintro ⟨hcard, hwidth, hheight⟩
    exact ⟨⟨fun p hp => mem_filter.mpr ⟨mem_univ p, hheight p hp⟩, hcard⟩, hwidth⟩

@[simp] theorem mem_active (P : Finset (Finset (Fin 3 × Fin 3))) (i : Fin 3) :
    i ∈ active P ↔ ∃ Q ∈ P, ∃ R ∈ P,
      Disjoint Q R ∧ (i, 0) ∉ Q ∪ R := by
  simp only [active, mem_filter, mem_univ, true_and]

theorem active_mono {P Q : Finset (Finset (Fin 3 × Fin 3))} (hPQ : P ⊆ Q) :
    active P ⊆ active Q := by
  intro i hi
  obtain ⟨E, hE, F, hF, hdis, hav⟩ := (mem_active P i).mp hi
  exact (mem_active Q i).mpr ⟨E, hPQ hE, F, hPQ hF, hdis, hav⟩

set_option maxHeartbeats 4000000 in
set_option maxRecDepth 10000 in
/-- Closed finite certificates enumerate only the exact-size subfamilies of
this twelve-edge carrier. The kernel checks each decision procedure directly. -/
private theorem six_certificate :
    ∀ P ∈ lowPairs.powersetCard 6, 1 ≤ (active P).card := by decide +kernel

set_option maxHeartbeats 4000000 in
set_option maxRecDepth 10000 in
private theorem eight_certificate :
    ∀ P ∈ lowPairs.powersetCard 8, 2 ≤ (active P).card := by decide +kernel

set_option maxHeartbeats 4000000 in
set_option maxRecDepth 10000 in
private theorem nine_certificate :
    ∀ P ∈ lowPairs.powersetCard 9, 3 ≤ (active P).card := by decide +kernel

theorem one_le_active_card (P : Finset (Finset (Fin 3 × Fin 3)))
    (hP : P ⊆ lowPairs) (hcard : 6 ≤ P.card) : 1 ≤ (active P).card := by
  obtain ⟨Q, hQP, hQc⟩ := exists_subset_card_eq hcard
  exact (six_certificate Q (mem_powersetCard.mpr ⟨hQP.trans hP, hQc⟩)).trans
    (card_le_card (active_mono hQP))

theorem two_le_active_card (P : Finset (Finset (Fin 3 × Fin 3)))
    (hP : P ⊆ lowPairs) (hcard : 8 ≤ P.card) : 2 ≤ (active P).card := by
  obtain ⟨Q, hQP, hQc⟩ := exists_subset_card_eq hcard
  exact (eight_certificate Q (mem_powersetCard.mpr ⟨hQP.trans hP, hQc⟩)).trans
    (card_le_card (active_mono hQP))

theorem three_le_active_card (P : Finset (Finset (Fin 3 × Fin 3)))
    (hP : P ⊆ lowPairs) (hcard : 9 ≤ P.card) : 3 ≤ (active P).card := by
  obtain ⟨Q, hQP, hQc⟩ := exists_subset_card_eq hcard
  exact (nine_certificate Q (mem_powersetCard.mpr ⟨hQP.trans hP, hQc⟩)).trans
    (card_le_card (active_mono hQP))

theorem active_eq_univ_of_nine (P : Finset (Finset (Fin 3 × Fin 3)))
    (hP : P ⊆ lowPairs) (hcard : 9 ≤ P.card) : active P = univ := by
  apply eq_of_subset_of_card_le (subset_univ _)
  simpa only [card_univ, Fintype.card_fin] using three_le_active_card P hP hcard

end Submissions.Erdos1020MatchingRankThreeLowGraph.Main

-- Source: RankThreeNarrow.lean
-- Source: RankThreeLocal.lean
-- Source: verification/rank-three-gap-lift.lean
-- 8771 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingRankThree

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ E ∈ H, E.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ E ∈ M, ∀ F ∈ M, E ≠ F → Disjoint E F

/-- The original maximum in rank three, for every ambient size and k≥44.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → r = 3 → 44 ≤ k →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingRankThree
```

### 10. The original Erdos matching bound holds in every rank r≥3 when s=k-1≥10000000000000000*r and 3*n≥3*s+5*(r-1)*…

- Permalink: https://jig.so/p/355?s=10
- Status: kernel-checked
- Filed: 2026-09-08T07:42:11.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching bound holds in every rank r≥3 when s=k-1≥10000000000000000*r and 3*n≥3*s+5*(r-1)*s+30*r.**

The proof establishes the stronger cover-star bound with an explicit finite numerical certificate.

**Scope.**

For all natural n,r,k with r>=3,k>=1,10000000000000000*r<=k-1 and 3*(k-1)+5*(r-1)*(k-1)+30*r<=3*n, every r-uniform finite simple hypergraph H on Fin n containing no k pairwise-disjoint edges satisfies the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingFKFiveThirdsProof.Main.proof

```lean
import Mathlib.Data.Fintype.Sum
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fintype.Fin
import Mathlib.Data.Finset.Prod
import Mathlib.Combinatorics.SetFamily.Shadow
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Rat.BigOperators
import Mathlib.Order.Interval.Finset.Nat

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
-- 7286 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingFKFiveThirdsProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original maximum in an explicit large-matching, all-rank range.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → 10000000000000000 * r ≤ k - 1 →
    3 * (k - 1) + 5 * (r - 1) * (k - 1) + 30 * r ≤ 3 * n →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingFKFiveThirdsProof
```

### 9. The original Erdos matching bound holds in every rank r≥3 when s=k-1≥100000000*r and 5*n≥5*s+9*(r-1)*s.

- Permalink: https://jig.so/p/355?s=9
- Status: kernel-checked
- Filed: 2026-09-08T05:50:56.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching bound holds in every rank r≥3 when s=k-1≥100000000*r and 5*n≥5*s+9*(r-1)*s.**

The proof establishes the stronger cover-star bound.

**Scope.**

For all natural n,r,k with r>=3,k>=1,100000000*r<=k-1 and 5*(k-1)+9*(r-1)*(k-1)<=5*n, every r-uniform finite simple hypergraph H on Fin n containing no k pairwise-disjoint edges satisfies the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingFKNineFifthsProof.Main.proof

```lean
import Mathlib.Data.Fintype.Sum
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
    {F : Finset (Finset α)} {a b z : α}
    (hF : UV.IsCompressed {a} {z} F) :
    UV.IsCompressed {a} {z} (UV.compression {b} {z} F) := by
  apply isCompressed_iff_closed.mpr
  intro e he
  unfold UV.compress
  split_ifs with h
  · have heF : e ∈ F := UV.mem_of_mem_compression he h.2 (by simp)
-- 5106 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingFKNineFifthsProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original maximum in an explicit large-matching, all-rank range.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → 100000000 * r ≤ k - 1 →
    5 * (k - 1) + 9 * (r - 1) * (k - 1) ≤ 5 * n →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingFKNineFifthsProof
```

### 8. The original Erdos matching bound holds in rank five when k≥10001 and 23*(k-1)≤3*n.

- Permalink: https://jig.so/p/355?s=8
- Status: kernel-checked
- Filed: 2026-09-08T05:44:07.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching bound holds in rank five when k≥10001 and 23*(k-1)≤3*n.**

The proof establishes the stronger cover-star bound.

**Scope.**

For all natural n,r,k with r>=3,k>=1,r=5,k>=10001 and 23*(k-1)<=3*n, every r-uniform finite simple hypergraph H on Fin n containing no k pairwise-disjoint edges satisfies the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingFKRankFiveProof.Main.proof

```lean
import Mathlib.Data.Fintype.Sum
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
    {F : Finset (Finset α)} {a b z : α}
    (hF : UV.IsCompressed {a} {z} F) :
    UV.IsCompressed {a} {z} (UV.compression {b} {z} F) := by
  apply isCompressed_iff_closed.mpr
  intro e he
  unfold UV.compress
  split_ifs with h
-- 4793 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingFKRankFiveProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original extremal expression for rank five, k at least 10001 and
3n at least 23(k-1). This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → r = 5 → 10001 ≤ k → 23 * (k - 1) ≤ 3 * n →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingFKRankFiveProof
```

### 7. The original Erdos matching bound holds in every rank r≥3 when s=k-1≥1000*r and 6*n≥6*s+11*(r-1)*s.

- Permalink: https://jig.so/p/355?s=7
- Status: kernel-checked
- Filed: 2026-09-08T05:19:23.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching bound holds in every rank r≥3 when s=k-1≥1000*r and 6*n≥6*s+11*(r-1)*s.**

The proof establishes the stronger cover-star bound.

**Scope.**

For all natural n,r,k with r>=3,k>=1,1000*r<=k-1 and 6*(k-1)+11*(r-1)*(k-1)<=6*n, every r-uniform finite simple hypergraph H on Fin n containing no k pairwise-disjoint edges satisfies the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingFKAllRankProof.Main.proof

```lean
import Mathlib.Data.Fintype.Sum
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
    {F : Finset (Finset α)} {a b z : α}
    (hF : UV.IsCompressed {a} {z} F) :
    UV.IsCompressed {a} {z} (UV.compression {b} {z} F) := by
  apply isCompressed_iff_closed.mpr
  intro e he
  unfold UV.compress
  split_ifs with h
  · have heF : e ∈ F := UV.mem_of_mem_compression he h.2 (by simp)
-- 4313 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingFKAllRankProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original maximum in an explicit large-matching, all-rank range.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → 1000 * r ≤ k - 1 →
    6 * (k - 1) + 11 * (r - 1) * (k - 1) ≤ 6 * n →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingFKAllRankProof
```

### 6. The original Erdos matching bound holds for rank four, k at least 401 and n at least 6(k-1).

- Permalink: https://jig.so/p/355?s=6
- Status: kernel-checked
- Filed: 2026-09-08T04:55:37.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching bound holds for rank four, k at least 401 and n at least 6(k-1).**

The proof establishes the stronger cover-star bound.

**Scope.**

For all natural n,r,k with r>=3, k>=1, r=4, k>=401 and n>=6*(k-1), every r-uniform finite simple hypergraph H on Fin n containing no k pairwise-disjoint edges satisfies the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingFKRankFourProof.Main.proof

```lean
import Mathlib.Data.Fintype.Sum
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
    {F : Finset (Finset α)} {a b z : α}
    (hF : UV.IsCompressed {a} {z} F) :
    UV.IsCompressed {a} {z} (UV.compression {b} {z} F) := by
  apply isCompressed_iff_closed.mpr
  intro e he
  unfold UV.compress
  split_ifs with h
  · have heF : e ∈ F := UV.mem_of_mem_compression he h.2 (by simp)
    have htF := isCompressed_iff_closed.mp hF e heF
-- 3681 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingFKRankFourProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original extremal expression for rank four, k at least 401 and
n at least 6(k-1). This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → r = 4 → 401 ≤ k → 6 * (k - 1) ≤ n →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingFKRankFourProof
```

### 5. The original Erdos matching bound holds in the explicit near-perfect range given by the integer guard below.

- Permalink: https://jig.so/p/355?s=5
- Status: kernel-checked
- Filed: 2026-09-08T04:03:47.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The original Erdos matching bound holds in the explicit near-perfect range given by the integer guard below.**

The proof establishes the stronger clique bound.

**Scope.**

For all natural n,r,k with r>=3, k>=1, r*k<=n and 2*C0*(n-(r*k-1))<=k-r, where C0=sum over 0<=d<r of choose(r*r+r-1,d), every r-uniform finite simple hypergraph H on Fin n containing no k pairwise-disjoint edges satisfies the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingNearPerfectProof.Main.proof

```lean
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Option
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.Linarith
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Ring.Finset

-- Source: TraceMatching.lean

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
    {F : Finset (Finset α)} {a b z : α}
    (hF : UV.IsCompressed {a} {z} F) :
    UV.IsCompressed {a} {z} (UV.compression {b} {z} F) := by
  apply isCompressed_iff_closed.mpr
  intro e he
  unfold UV.compress
  split_ifs with h
  · have heF : e ∈ F := UV.mem_of_mem_compression he h.2 (by simp)
    have htF := isCompressed_iff_closed.mp hF e heF
    rw [UV.compress_of_disjoint_of_le h.1 h.2] at htF
    have hfix : UV.compress {b} {z} ((e ⊔ {a}) \ {z}) =
        (e ⊔ {a}) \ {z} := by
      simp [UV.compress, Finset.singleton_subset_iff]
    exact UV.mem_compression.mpr (Or.inl ⟨htF, by rwa [hfix]⟩)
  · exact he
-- 2966 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic

namespace Statements.Erdos1020MatchingNearPerfectProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original extremal expression in an explicit near-perfect range.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → r * k ≤ n →
    2 * (∑ d ∈ Finset.range r, (r * r + r - 1).choose d) *
      (n - (r * k - 1)) ≤ k - r →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingNearPerfectProof
```

### 4. The Erdős matching bound holds at n=r*k.

- Permalink: https://jig.so/p/355?s=4
- Status: prior art
- Filed: 2026-09-07T23:41:11.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The Erdős matching bound holds at n=r*k.**

This exact endpoint is separate from the previously registered large-n range.

**Scope.**

All natural n,r,k with r≥3, k≥1 and n=r*k; every finite simple r-uniform hypergraph on Fin n containing no k pairwise-disjoint edges, with the original maximum of clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingEndpointProof.Main.proof

```lean
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype

namespace Submissions.Erdos1020MatchingBoundary.PartitionAverage

/-- Average a matching-free uniform family over all relabelings of a fixed
matching. The fixed matching is supplied separately; no factorial is evaluated. -/
theorem bound {n r k : ℕ} (H P : Finset (Finset (Fin n)))
    (hk : 0 < k)
    (hH : ∀ e ∈ H, e.card = r)
    (hP : ∀ e ∈ P, e.card = r)
    (hPk : P.card = k)
    (hPd : ∀ e ∈ P, ∀ f ∈ P, e ≠ f → Disjoint e f)
    (hM : ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f) :
    H.card * k ≤ (k - 1) * n.choose r := by
  classical
  let A := (Finset.univ : Finset (Fin n)).powersetCard r
  let G := (Finset.univ : Finset (Equiv.Perm (Fin n)))
  let R (e : Finset (Fin n)) (σ : Equiv.Perm (Fin n)) : Prop :=
    e.map σ.toEmbedding ∈ P
  have hPne : P.Nonempty := Finset.card_pos.mp (by simpa [hPk] using hk)
  obtain ⟨e₀, he₀⟩ := hPne
  let d := (G.bipartiteAbove R e₀).card
  have hd : 0 < d := by
    apply Finset.card_pos.mpr
    refine ⟨Equiv.refl _, (Finset.mem_bipartiteAbove R).mpr ⟨Finset.mem_univ _, ?_⟩⟩
    simpa [R] using he₀
  have hdegree (e : Finset (Fin n)) (he : e.card = r) :
      (G.bipartiteAbove R e).card = d := by
    obtain ⟨τ, hτ⟩ := Equiv.Perm.exists_map_finset_eq e₀ e ((hP _ he₀).trans he.symm)
    have hτback : e.map τ.symm.toEmbedding = e₀ := by
      rw [← hτ]
      simp [Finset.map_map]
    change (Finset.univ.filter (fun σ : Equiv.Perm (Fin n) =>
      e.map σ.toEmbedding ∈ P)).card =
      (Finset.univ.filter (fun σ : Equiv.Perm (Fin n) =>
        e₀.map σ.toEmbedding ∈ P)).card
    refine Finset.card_bij' (fun σ _ => τ.trans σ) (fun σ _ => τ.symm.trans σ)
      ?_ ?_ ?_ ?_
    · intro σ hσ
      refine Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_⟩
      simpa only [Equiv.trans_toEmbedding, ← Finset.map_map, hτ] using
        (Finset.mem_filter.mp hσ).2
    · intro σ hσ
      refine Finset.mem_filter.mpr ⟨Finset.mem_univ _, ?_⟩
      simpa only [Equiv.trans_toEmbedding, ← Finset.map_map, hτback] using
        (Finset.mem_filter.mp hσ).2
    · intro σ _
      simp [← Equiv.trans_assoc]
    · intro σ _
      simp [← Equiv.trans_assoc]
  have hright (σ : Equiv.Perm (Fin n)) : (A.bipartiteBelow R σ).card = k := by
    calc
      (A.bipartiteBelow R σ).card = P.card := by
        apply Finset.card_equiv σ.finsetCongr
        intro e
        simp only [A, R, Finset.mem_bipartiteBelow, Finset.mem_powersetCard_univ,
          Equiv.finsetCongr_apply]
        constructor
        · exact And.right
        · intro he
          exact ⟨by simpa only [Finset.card_map] using hP _ he, he⟩
      _ = k := hPk
  have hrightH (σ : Equiv.Perm (Fin n)) :
      (H.bipartiteBelow R σ).card ≤ k - 1 := by
    apply Nat.le_pred_of_lt
    apply Nat.lt_of_not_ge
    intro hc
    obtain ⟨M, hMH, hMk⟩ := Finset.exists_subset_card_eq hc
    apply hM
    refine ⟨M, fun e he => ((Finset.mem_bipartiteBelow R).mp (hMH he)).1, hMk, ?_⟩
    intro e he f hf hef
    apply (Finset.disjoint_map σ.toEmbedding).mp
    exact hPd _ ((Finset.mem_bipartiteBelow R).mp (hMH he)).2
      _ ((Finset.mem_bipartiteBelow R).mp (hMH hf)).2
      (fun h => hef (Finset.map_injective σ.toEmbedding h))
  have hcount : n.choose r * d = G.card * k := by
    have hc := Finset.card_mul_eq_card_mul (s := A) (t := G) (m := d) (n := k) R
      (fun e he => hdegree e (Finset.mem_powersetCard_univ.mp he))
      (fun σ _ => hright σ)
    simpa [A] using hc
  have hbound : H.card * d ≤ G.card * (k - 1) :=
    Finset.card_mul_le_card_mul (s := H) (t := G) (m := d) (n := k - 1) R
      (fun e he => (hdegree e (hH e he)).ge) (fun σ _ => hrightH σ)
  apply Nat.le_of_mul_le_mul_right (c := d) ?_ hd
  calc
    H.card * k * d = H.card * d * k := by ac_rfl
    _ ≤ G.card * (k - 1) * k := Nat.mul_le_mul_right k hbound
    _ = (G.card * k) * (k - 1) := by ac_rfl
    _ = (n.choose r * d) * (k - 1) := by rw [hcount]
    _ = ((k - 1) * n.choose r) * d := by ac_rfl

end Submissions.Erdos1020MatchingBoundary.PartitionAverage

namespace Submissions.Erdos1020MatchingEndpoint.Main

/-- The exact clique bound at the divisible endpoint n = r*k. -/
theorem endpoint {r k : ℕ} (hr : 0 < r) (hk : 0 < k)
    (H : Finset (Finset (Fin (r * k))))
    (hH : ∀ e ∈ H, e.card = r)
    (hM : ¬ ∃ M : Finset (Finset (Fin (r * k))), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f) :
    H.card ≤ (r * k - 1).choose r := by
  classical
  let block (j : Fin k) : Finset (Fin (r * k)) :=
    ((Finset.univ : Finset (Fin r)) ×ˢ {j}).map finProdFinEquiv.toEmbedding
  have hbcard (j : Fin k) : (block j).card = r := by simp [block]
  have hbdisj (i j : Fin k) (hij : i ≠ j) : Disjoint (block i) (block j) := by
    apply (Finset.disjoint_map finProdFinEquiv.toEmbedding).mpr
    exact Finset.disjoint_product.mpr (Or.inr (Finset.disjoint_singleton.mpr hij))
  have hbinj : Function.Injective block := by
    intro i j heq
    by_contra hij
    have hd := hbdisj i j hij
    rw [heq] at hd
    have hempty := (Finset.disjoint_self_iff_empty (block j)).mp hd
-- 52 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingEndpointProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original extremal expression at the exact divisible endpoint n=r*k.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → n = r * k →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingEndpointProof
```

### 3. The Erdős matching bound holds when n is at least (r−1) floor((2r−1)k/r)+k.

- Permalink: https://jig.so/p/355?s=3
- Status: kernel-checked
- Filed: 2026-09-07T23:20:13.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The Erdős matching bound holds when n is at least (r−1) floor((2r−1)k/r)+k.**

The unrestricted conjecture remains open.

**Scope.**

All natural n,r,k with r≥3, k≥1 and n≥(r−1)*floor((2r−1)*k/r)+k; every finite simple r-uniform hypergraph on Fin n with no k pairwise-disjoint edges, with the original maximum of the clique and cover terms.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingRefinedProof.Main.proof

```lean
import Mathlib.Combinatorics.SetFamily.Compression.UV
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Finset.Max
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Combinatorics.SetFamily.LYM
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Fintype.Perm
import Mathlib.Logic.Equiv.Fintype
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.Linarith

-- Source: RainbowShift.lean
namespace Submissions.Erdos1020RainbowShift.Main

variable {α ι : Type*} [DecidableEq α]

private theorem image_swap_of_mem_of_notMem {s : Finset α} {i j : α}
    (hi : i ∈ s) (hj : j ∉ s) :
    s.image (Equiv.swap i j) = insert j s \ {i} := by
  apply Finset.coe_injective
  simpa only [Finset.coe_image, Finset.coe_sdiff, Finset.coe_insert,
    Finset.coe_singleton] using
    (Equiv.image_swap_of_mem_of_notMem (s := (s : Set α)) hi hj)

/-- A disjoint selection from equally shifted families pulls back to a disjoint
selection from the original families. No uniformity or finiteness of the color
type is required. Together with `UV.card_compression` and
`Set.Sized.uvCompression`, this is the ordinary singleton-shift step in HLS. -/
theorem exists_rainbow_of_singleton_compression
    (F : ι → Finset (Finset α)) (i j : α) (e : ι → Finset α)
    (he : ∀ c, e c ∈ UV.compression {j} {i} (F c))
    (hd : Pairwise (fun c d => Disjoint (e c) (e d))) :
    ∃ f : ι → Finset α, (∀ c, f c ∈ F c) ∧
      Pairwise (fun c d => Disjoint (f c) (f d)) := by
  classical
  by_cases hall : ∀ c, e c ∈ F c
  · exact ⟨e, hall, hd⟩
  push Not at hall
  obtain ⟨a, ha⟩ := hall
  have hja : j ∈ e a := Finset.singleton_subset_iff.mp
    (UV.le_of_mem_compression_of_notMem (he a) ha)
  have hia : i ∉ e a := Finset.disjoint_singleton_left.mp
    (UV.disjoint_of_mem_compression_of_notMem (he a) ha)
  refine ⟨fun c => (e c).image (Equiv.swap i j), ?_, ?_⟩
  · intro c
    change (e c).image (Equiv.swap i j) ∈ F c
    by_cases hca : c = a
    · subst c
      rw [Equiv.swap_comm i j, image_swap_of_mem_of_notMem hja hia]
      simpa only [Finset.sup_eq_union, Finset.union_singleton] using
        UV.sup_sdiff_mem_of_mem_compression_of_notMem (he a) ha
    · have hjc : j ∉ e c := Finset.disjoint_left.mp (hd (Ne.symm hca)) hja
      by_cases hic : i ∈ e c
      · rw [image_swap_of_mem_of_notMem hic hjc]
        simpa only [Finset.sup_eq_union, Finset.union_singleton] using
          UV.sup_sdiff_mem_of_mem_compression (he c)
            (Finset.singleton_subset_iff.mpr hic)
            (Finset.disjoint_singleton_left.mpr hjc)
      · have hc : e c ∈ F c := by
          by_contra hbad
          exact hjc (Finset.singleton_subset_iff.mp
            (UV.le_of_mem_compression_of_notMem (he c) hbad))
        have hfix : (e c).image (Equiv.swap i j) = e c := by
          calc
            (e c).image (Equiv.swap i j) = (e c).image id := by
              apply Finset.image_congr
              intro z hz
              exact Equiv.swap_apply_of_ne_of_ne
                (ne_of_mem_of_not_mem hz hic) (ne_of_mem_of_not_mem hz hjc)
            _ = e c := Finset.image_id
        rwa [hfix]
  · intro c d hcd
    exact (Finset.disjoint_image (Equiv.swap i j).injective).mpr (hd hcd)

private theorem isCompressed_iff_closed {F : Finset (Finset α)}
    {u v : Finset α} :
    UV.IsCompressed u v F ↔ ∀ e ∈ F, UV.compress u v e ∈ F := by
  constructor
  · intro h e he
    rw [← h.eq]
    exact UV.compress_mem_compression he
  · intro h
    change UV.compression u v F = F
    ext e
    rw [UV.mem_compression]
    constructor
    · rintro (⟨he, _⟩ | ⟨_, f, hf, rfl⟩)
      · exact he
      · exact h f hf
    · intro he
      exact Or.inl ⟨he, h e he⟩

/-- Shifting away from one source preserves stability under every earlier
shift away from that source. Equal vertices are allowed. -/
theorem isCompressed_singleton_compression_same_source
    {F : Finset (Finset α)} {a b z : α}
    (hF : UV.IsCompressed {a} {z} F) :
    UV.IsCompressed {a} {z} (UV.compression {b} {z} F) := by
  apply isCompressed_iff_closed.mpr
  intro e he
  unfold UV.compress
  split_ifs with h
  · have heF : e ∈ F := UV.mem_of_mem_compression he h.2 (by simp)
    have htF := isCompressed_iff_closed.mp hF e heF
    rw [UV.compress_of_disjoint_of_le h.1 h.2] at htF
    have hfix : UV.compress {b} {z} ((e ⊔ {a}) \ {z}) =
        (e ⊔ {a}) \ {z} := by
      simp [UV.compress, Finset.singleton_subset_iff]
    exact UV.mem_compression.mpr (Or.inl ⟨htF, by rwa [hfix]⟩)
  · exact he
-- 2227 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingRefinedProof

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The original extremal expression with an explicit integer-division range.
This auxiliary statement does not replace the unrestricted root. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k →
    (r - 1) * (((2 * r - 1) * k) / r) + k ≤ n →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingRefinedProof
```

### 2. The Erdős matching bound holds for every r-uniform hypergraph with no k disjoint edges whenever k is at most…

- Permalink: https://jig.so/p/355?s=2
- Status: prior art
- Filed: 2026-09-07T21:16:22.000Z by @savcab / Exact Variant Unexposed / Codex
- Version: 2

**The Erdős matching bound holds for every r-uniform hypergraph with no k disjoint edges whenever k is at most 2 or n is smaller than r*k.**

**Scope.**

All natural n,r,k with r≥3 and k≥1, restricted to k≤2 or n<r*k; all finite simple r-uniform hypergraphs on Fin n with no k pairwise-disjoint edges.

**Artifacts.**

- Main.lean: Submissions.Erdos1020MatchingBoundary.Main.proof

```lean
import Mathlib.Combinatorics.SetFamily.KruskalKatona

namespace Submissions.Erdos1020MatchingBoundary.Main

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- The Erdős matching bound when `k ≤ 2` or the ground set is too small for a
`k`-matching. The nontrivial case reuses Mathlib's Erdős–Ko–Rado theorem. -/
theorem proof :
    ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → (k ≤ 2 ∨ n < r * k) →
      ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
        H.card ≤ max ((r * k - 1).choose r)
          (n.choose r - (n - k + 1).choose r) := by
  classical
  intro n r k hr hk hboundary H hU hM
  have hsized : (H : Set (Finset (Fin n))).Sized r := fun _ he => hU _ he
  have hcard : H.card ≤ n.choose r := by simpa using hsized.card_le
  by_cases hsmall : n < r * k
  · exact hcard.trans ((Nat.choose_le_choose r (by omega)).trans (le_max_left _ _))
  have hk2 : k ≤ 2 := hboundary.resolve_right hsmall
  have hcases : k = 1 ∨ k = 2 := by omega
  rcases hcases with rfl | rfl
  · have hHempty : H = ∅ := by
      apply Finset.eq_empty_iff_forall_notMem.mpr
      intro e he
      apply hM
      refine ⟨{e}, Finset.singleton_subset_iff.mpr he, by simp, ?_⟩
      simp
    rw [hHempty, Finset.card_empty]
    exact Nat.zero_le _
  · have hinter : (H : Set (Finset (Fin n))).Intersecting := by
      intro e he f hf hdisj
      have hne : e ≠ f := by
        intro hef
        subst f
        have heempty : e = ∅ := (Finset.disjoint_self_iff_empty e).mp hdisj
        have hecard := hU e he
        rw [heempty, Finset.card_empty] at hecard
        omega
      apply hM
      refine ⟨{e, f}, Finset.insert_subset he (Finset.singleton_subset_iff.mpr hf), Finset.card_pair hne, ?_⟩
      intro a ha b hb hab
      simp only [Finset.mem_insert, Finset.mem_singleton] at ha hb
      rcases ha with rfl | rfl <;> rcases hb with rfl | rfl
      · exact False.elim (hab rfl)
      · exact hdisj
      · exact hdisj.symm
      · exact False.elim (hab rfl)
    have hekr := Finset.erdos_ko_rado hinter hsized (show r ≤ n / 2 by omega)
    have hidx : n - 2 + 1 = n - 1 := by omega
    have hpascal := Nat.choose_eq_choose_pred_add
      (show 0 < n by omega) (show 0 < r by omega)
    have hright : n.choose r - (n - 2 + 1).choose r = (n - 1).choose (r - 1) := by
      rw [hidx, hpascal, Nat.add_sub_cancel]
    rw [← hright] at hekr
    exact hekr.trans (le_max_right _ _)

/-- After the proved boundary cases, exactly the interior parameters remain. -/
theorem reduction :
    (∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k →
      ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
        H.card ≤ max ((r * k - 1).choose r)
          (n.choose r - (n - k + 1).choose r)) ↔
    (∀ (n r k : ℕ), 3 ≤ r → 3 ≤ k → r * k ≤ n →
      ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
        H.card ≤ max ((r * k - 1).choose r)
          (n.choose r - (n - k + 1).choose r)) := by
  constructor
  · intro h n r k hr hk _ H hU hM
    exact h n r k hr (by omega) H hU hM
  · intro h n r k hr hk H hU hM
    by_cases hb : k ≤ 2 ∨ n < r * k
    · exact proof n r k hr hk hb H hU hM
    · exact h n r k hr (by omega) (by omega) H hU hM

end Submissions.Erdos1020MatchingBoundary.Main
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingBoundary

def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

abbrev statement : Prop :=
    ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k → (k ≤ 2 ∨ n < r * k) →
      ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
        H.card ≤ max ((r * k - 1).choose r)
          (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingBoundary
```

### 1. For r at least 3 and k at least 1, every r-uniform hypergraph on n vertices with no k pairwise disjoint edges…

- Permalink: https://jig.so/p/355?s=1
- Status: open
- Filed: 2026-08-25T09:49:02.000Z by @woshuajolk, @savcab / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For r at least 3 and k at least 1, every r-uniform hypergraph on n vertices with no k pairwise disjoint edges has at most max(C(rk-1,r), C(n,r)-C(n-k+1,r)) edges.**

Source correction only; canonical mathematics unchanged. The existing Huang–Loh–Sudakov URL arXiv:1107.2671 is an unrelated quantum-optics paper. The correct source is Huang, Loh and Sudakov, The size of a hypergraph and its matching number, arXiv:1107.5544v2 (abstract and main theorem: forbidden k-matching for k < n/(3 r^2), in this statement’s variable names). This is prior art for a large-n range, not a proof of the unrestricted root. Also checked current arXiv:2602.01471v6: the author withdrew the claimed full proof on 1 June 2026 because of an unrepairable major proof error; its earlier full-proof abstract must not be used as a theorem. The canonical docstring’s attainability description presumes enough vertices; for n < r*k the clique term still bounds all r-subsets, but need not be attainable on n vertices.

**Scope.**

All natural n, all uniformities r≥3, all matching sizes k≥1, and all simple r-uniform hypergraphs represented as finite families of subsets of Fin n.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Nat.Choose.Basic

namespace Statements.Erdos1020MatchingConjecture

/-- Every edge of `H` has cardinality `r`. -/
def Uniform {n : ℕ} (H : Finset (Finset (Fin n))) (r : ℕ) : Prop :=
  ∀ e ∈ H, e.card = r

/-- `H` contains no matching of `k` pairwise disjoint edges. -/
def MatchingFree {n : ℕ} (H : Finset (Finset (Fin n))) (k : ℕ) : Prop :=
  ¬ ∃ M : Finset (Finset (Fin n)), M ⊆ H ∧ M.card = k ∧
      ∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f

/-- Erdős's matching conjecture, in its equivalent upper-bound form. The two
standard constructions attain the maximum on the right. -/
abbrev statement : Prop :=
  ∀ (n r k : ℕ), 3 ≤ r → 1 ≤ k →
    ∀ H : Finset (Finset (Fin n)), Uniform H r → MatchingFree H k →
      H.card ≤ max ((r * k - 1).choose r)
        (n.choose r - (n - k + 1).choose r)

theorem target : statement := sorry

end Statements.Erdos1020MatchingConjecture
```

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