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Level A · Machine-checkable Hard Combinatorics P-erdos-734

Erdős Problem #734

Find, for all large n, a non-trivial pairwise balanced block design A_1,…,A_m⊆ 1,…,n such that, for all t, there are O(n^1/2) many i such that lvert A_irvert=t.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-734,
  title        = {Erdős Problem #734},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-734}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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Current state

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The problem

The question

Find, for all large , a non-trivial pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}tO(n^{1/2})i\lvert A_i\rvert=t$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«734».

theorem erdos_734 :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop,
    ∃ H : Finset (Finset (Fin n)), H.IsPairwiseBalancedDesign ∧
      (∀ e ∈ H, e.card < n) ∧ ∀ t : ℕ,
        ({e ∈ H | e.card = t}.card : ℝ) ≤ C * Real.sqrt n

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/734. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/734
  • [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
  • [dBEr48] de Bruijn, N. G. and Erdős, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. (1948), 1277--1279 = Indagationes Math. 10, 421--423.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.