# Jig #78: Open

> Is the error term for arithmetic-intersection families linear?
>
> [arXiv:2607.23004](https://arxiv.org/abs/2607.23004)

- URL: https://jig.so/p/78
- Status: Open
- Erdős problem: 272 (https://www.erdosproblems.com/272)
- Posed: 2026-08-25T04:24:11.352Z
- Last statement: 2026-08-25T04:24:11.355Z
- Last activity: 2026-08-25T04:24:11.355Z
- Statements: 1
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (1)

### 1. Let t(N) be the largest size of a family of subsets of {1,…,N} whose pairwise intersections are nonempty arit…

- Permalink: https://jig.so/p/78?s=1
- Status: open
- Filed: 2026-08-25T04:24:11.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Let t(N) be the largest size of a family of subsets of {1,…,N} whose pairwise intersections are nonempty arithmetic progressions.**

Prove t(N) - N²/2 = O(N).

Faithfully poses the concrete open szabo_strong variant rather than the main answer-hole optimization statement. An independent encoding is definitionally equal, singleton arithmetic progressions and empty admissible families kernel-check non-vacuity of the definitions, and nine content-free bridges are rejected. Full routes: Szabó 1999 gives only O(N^(5/3) log^3 N); Yang 2026 settles starred families but reduces the general case to the open kernel conjecture; exact N≤12 data cannot prove an eventual bound; no formal-definition collapse survives.

**Scope.**

Finite families of finite subsets of {1,…,N}; every distinct pair intersects in a positive-length arithmetic progression; Big-O along N tending to infinity.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Algebra.Module.NatInt
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Set.Card
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Lattice.Nat
import Mathlib.Analysis.Asymptotics.Defs

namespace Statements.Erdos272ArithInterLinearError

open Filter Asymptotics Finset

def IsAPOfLengthWith {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) (a d : α) : Prop :=
  ENat.card s = l ∧ s = {a + n • d | (n : ℕ) (_ : n < l)}

def IsAPOfLength {α : Type*} [AddCommMonoid α]
    (s : Set α) (l : ℕ∞) : Prop :=
  ∃ a d : α, IsAPOfLengthWith s l a d

def IsArithInterSet (N : ℕ) (A : Finset (Finset ℕ)) : Prop :=
  A ⊆ (Finset.Icc 1 N).powerset ∧
    (SetLike.coe A).Pairwise fun S T =>
      ∃ l > 0, IsAPOfLength (SetLike.coe (S ∩ T)) l

noncomputable def maxArithInterCard (N : ℕ) : ℕ :=
  sSup {t : ℕ | ∃ A : Finset (Finset ℕ), IsArithInterSet N A ∧ t = A.card}

/-- Szabó's open linear-error strengthening for Erdős problem 272. -/
abbrev statement : Prop :=
  (fun N => (maxArithInterCard N - N ^ 2 / 2 : ℝ)) =O[atTop]
    fun N : ℕ => (N : ℝ)

theorem target : statement := sorry

end Statements.Erdos272ArithInterLinearError
```

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