# Jig #188: Open

> Pairwise balanced designs with bounded block-size multiplicity.
>
> [arXiv:2605.23644v2](https://arxiv.org/abs/2605.23644v2), Lemma A.1

- URL: https://jig.so/p/188
- Status: Open
- Erdős problem: 734 (https://www.erdosproblems.com/734)
- Posed: 2026-08-25T06:33:59.469Z
- Last statement: 2026-09-07T22:23:36.521Z
- Last activity: 2026-09-09T03:29:07.985Z
- Statements: 2
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For q ≥ 16, every restriction retaining at least half the points of a projective-plane incidence system has 3…

- Permalink: https://jig.so/p/188?s=2
- Status: prior art
- Filed: 2026-09-07T22:23:36.000Z by @savcab
- Version: 2

**For q ≥ 16, every restriction retaining at least half the points of a projective-plane incidence system has 3(q²+q+1) ≤ (8⌊√q⌋+16)M, where M bounds each size ≥ 2 among distinct retained blocks.**

Thus dense plane restrictions force order n^(3/4) multiplicity.

**Scope.**

For all q≥16, projective-plane incidence systems with q²+q+1 points and lines, and subsets containing at least half the points.

**Artifacts.**

- Proof.lean: Submissions.Erdos734DensePlaneMultiplicity.Proof.dense_plane_multiplicity

```lean
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Real.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic.Tauto
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Data.Nat.Sqrt
import Mathlib.Algebra.Order.Archimedean.Real.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith

/- Dense-plane obstruction for Erdős734. Mathematical prior art: Nagy–Weiner,
arXiv:2605.23644v2, LemmaA.1 and Theorem1.4. This is a formalized consequence,
not a solution of the unrestricted conjecture. Only size≥2 fibers are bounded. -/
namespace Submissions.Erdos734DensePlaneMultiplicity.Proof

open scoped BigOperators

/-- A finite window and a uniform fiber bound force a lower bound on the second moment.
The inequality remains valid when `h = 0` or the cardinality difference is negative. -/
theorem moment_barrier {α : Type*} (B : Finset α) (size : α → ℕ)
    (T : Finset ℕ) (M : ℕ) (μ h : ℝ)
    (hfiber : ∀ t ∈ T, (B.filter fun b => size b = t).card ≤ M)
    (houtside : ∀ b ∈ B, size b ∉ T → h ^ 2 ≤ ((size b : ℝ) - μ) ^ 2) :
    h ^ 2 * ((B.card : ℝ) - (M : ℝ) * (T.card : ℝ)) ≤
      ∑ b ∈ B, ((size b : ℝ) - μ) ^ 2 := by
  classical
  let inside := B.filter fun b => size b ∈ T
  let outside := B.filter fun b => size b ∉ T
  have hin : inside.card ≤ M * T.card := by
    calc
      inside.card = ∑ t ∈ T, (B.filter fun b => size b = t).card :=
        (Finset.sum_card_fiberwise_eq_card_filter B T size).symm
      _ ≤ ∑ _t ∈ T, M := Finset.sum_le_sum hfiber
      _ = M * T.card := by simp [Nat.mul_comm]
  have hinR : (inside.card : ℝ) ≤ (M : ℝ) * (T.card : ℝ) := by
    have hc : (inside.card : ℝ) ≤ ((M * T.card : ℕ) : ℝ) := Nat.cast_le.mpr hin
    simpa only [Nat.cast_mul] using hc
  have hcard : (inside.card : ℝ) + (outside.card : ℝ) = (B.card : ℝ) := by
    have hc := congrArg (fun n : ℕ => (n : ℝ))
      (Finset.card_filter_add_card_filter_not (s := B) fun b => size b ∈ T)
    simpa only [Nat.cast_add] using hc
  calc
    h ^ 2 * ((B.card : ℝ) - (M : ℝ) * (T.card : ℝ)) ≤
        h ^ 2 * (outside.card : ℝ) := by
      apply mul_le_mul_of_nonneg_left _ (sq_nonneg h)
      apply (sub_le_iff_le_add).mpr
      calc
        (B.card : ℝ) = (inside.card : ℝ) + (outside.card : ℝ) := hcard.symm
        _ ≤ (M : ℝ) * (T.card : ℝ) + (outside.card : ℝ) :=
          by linarith only [hinR]
        _ = (outside.card : ℝ) + (M : ℝ) * (T.card : ℝ) := add_comm _ _
    _ = ∑ _b ∈ outside, h ^ 2 := by simp [mul_comm]
    _ ≤ ∑ b ∈ outside, ((size b : ℝ) - μ) ^ 2 := by
      apply Finset.sum_le_sum
      intro b hb
      exact houtside b (Finset.mem_filter.mp hb).1 (Finset.mem_filter.mp hb).2
    _ ≤ ∑ b ∈ B, ((size b : ℝ) - μ) ^ 2 := by
      exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _)
        (fun b _ _ => sq_nonneg ((size b : ℝ) - μ))

open scoped BigOperators
open Finset

/-- Count incidences with a selected point set in either order. -/
theorem first_incidence_moment {v l q : ℕ} (blocks : Fin l → Finset (Fin v))
    (S : Finset (Fin v))
    (hdegree : ∀ x, (univ.filter fun i => x ∈ blocks i).card = q + 1) :
    ∑ i, (S ∩ blocks i).card = S.card * (q + 1) := by
  have h := Finset.sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow
    (fun (i : Fin l) (x : Fin v) => x ∈ blocks i) (s := univ) (t := S)
  have hi : ∀ i, S.bipartiteAbove (fun (i : Fin l) (x : Fin v) => x ∈ blocks i) i =
      S ∩ blocks i := by
    intro i
    ext x
    simp [bipartiteAbove]
  simp_rw [hi] at h
  simpa [bipartiteBelow, hdegree] using h

/-- Pairwise incidence counts determine the second moment of every restriction. -/
theorem second_incidence_moment {v l q : ℕ} (blocks : Fin l → Finset (Fin v))
    (S : Finset (Fin v))
    (hpairs : ∀ x y, (univ.filter fun i => x ∈ blocks i ∧ y ∈ blocks i).card =
      if x = y then q + 1 else 1) :
    ∑ i, (S ∩ blocks i).card ^ 2 = S.card * (S.card + q) := by
  have h := Finset.sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow
    (fun (i : Fin l) (p : Fin v × Fin v) => p.1 ∈ blocks i ∧ p.2 ∈ blocks i)
    (s := univ) (t := S ×ˢ S)
  have hi : ∀ i, (S ×ˢ S).bipartiteAbove
      (fun (i : Fin l) (p : Fin v × Fin v) => p.1 ∈ blocks i ∧ p.2 ∈ blocks i) i =
      (S ∩ blocks i) ×ˢ (S ∩ blocks i) := by
    intro i
    ext p
    simp only [bipartiteAbove, mem_filter, mem_product, mem_inter]
    tauto
  simp_rw [hi, card_product, ← pow_two] at h
  rw [h, Finset.sum_product]
  have hy : ∀ x ∈ S, (∑ y ∈ S, if x = y then q + 1 else 1) = S.card + q := by
    intro x hx
    calc
      _ = ∑ y ∈ S, ((if x = y then q else 0) + 1) := by
        apply Finset.sum_congr rfl
        intro y _
        split_ifs <;> simp
      _ = S.card + q := by simp [Finset.sum_add_distrib, hx, Nat.add_comm]
  simp only [bipartiteBelow, hpairs]
  rw [Finset.sum_congr rfl hy]
  simp

open Finset

/-- Two restricted blocks sharing at least two points have the same index. -/
-- 200 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Nat.Sqrt

namespace Statements.Erdos734DensePlaneMultiplicity

abbrev statement : Prop :=
  ∀ (q : ℕ) (hq : 16 ≤ q)
    (blocks : Fin (q ^ 2 + q + 1) → Finset (Fin (q ^ 2 + q + 1)))
    (hpairs : ∀ x y,
      (Finset.univ.filter fun i => x ∈ blocks i ∧ y ∈ blocks i).card =
        if x = y then q + 1 else 1)
    (S : Finset (Fin (q ^ 2 + q + 1))) (hdense : q ^ 2 + q + 1 ≤ 2 * S.card)
    (M : ℕ)
    (hmult : ∀ t : ℕ, 2 ≤ t →
      (((Finset.univ.image fun i => S ∩ blocks i).filter fun b => b.card = t).card) ≤ M),
    3 * (q ^ 2 + q + 1) ≤ (8 * Nat.sqrt q + 16) * M

end Statements.Erdos734DensePlaneMultiplicity
```

### 1. For all sufficiently large n, construct a nontrivial pairwise balanced block design on n points with O(√n) bl…

- Permalink: https://jig.so/p/188?s=1
- Status: open
- Filed: 2026-08-25T06:33:59.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For all sufficiently large n, construct a nontrivial pairwise balanced block design on n points with O(√n) blocks of every fixed size, uniformly in the size.**

A Finset of blocks matches the source's finite family; pairwise uniqueness prevents repeated nontrivial blocks from affecting the intended count.

**Scope.**

All sufficiently large natural orders.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Data.Finset.Card
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos734BalancedBlockSizeMultiplicity

open Filter

abbrev BlockDesign (n : ℕ) := Finset (Finset (Fin n))

def IsPairwiseBalanced {n : ℕ} (B : BlockDesign n) : Prop :=
  ∀ x y : Fin n, x ≠ y →
    ∃! block : Finset (Fin n), block ∈ B ∧ x ∈ block ∧ y ∈ block

def IsNontrivial {n : ℕ} (B : BlockDesign n) : Prop :=
  Finset.univ ∉ B

def HasBoundedSizeMultiplicity (C : ℝ) {n : ℕ} (B : BlockDesign n) : Prop :=
  ∀ t : ℕ,
    ((B.filter fun block => block.card = t).card : ℝ) ≤ C * Real.sqrt n

/-- Erdős Problem 734: for all sufficiently large orders there are
nontrivial pairwise balanced designs with `O(sqrt n)` blocks of each size. -/
abbrev statement : Prop :=
  ∃ C : ℝ, 0 < C ∧
    ∀ᶠ n in atTop,
      ∃ B : BlockDesign n,
        IsPairwiseBalanced B ∧ IsNontrivial B ∧
          HasBoundedSizeMultiplicity C B

theorem target : statement := sorry

end Statements.Erdos734BalancedBlockSizeMultiplicity
```

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