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Level A · Machine-checkable Hard Combinatorics P-erdos-724

Erdős Problem #724

Let f(n) be the maximum number of mutually orthogonal Latin squares of order n. Is it true that f(n) ≫ n^1/2? This question is from [Er81].

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

Current state (tracked automatically)

  • erdosproblems.com lists Erdős problem #724 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_724 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #724 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_724`research openFormal Conjectures
Formal Conjectures: research statements in the file1 open, 5 solvedFormal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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-724,
  title        = {Erdős Problem #724},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-724}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Disputed
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On the literature board
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Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. Paste its answer here — check it and publish it under your name

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.

or , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let be the maximum number of mutually orthogonal Latin squares of order . Is it true that This question is from [Er81].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«724».

theorem erdos_724 :
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, c * Real.sqrt (n : ℝ) ≤ (maxMOLS n : ℝ)

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/724. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/724
  • OEIS A001438
  • [Be83c] Beth, Thomas, *Eine Bemerkung zur Abschätzung der Anzahl orthogonaler lateinischer Quadrate mittels Siebverfahren*. Abh. Math. Sem. Univ. Hamburg (1983), 284-288.
  • [BPS60] Bose, R. C. and Shrikhande, S. S. and Parker, E. T., *Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture*. Canadian J. Math. (1960), 189-203.
  • [CES60] Chowla, S. and Erdős, P. and Straus, E. G., *On the maximal number of pairwise orthogonal Latin squares of a given order*. Canadian J. Math. (1960), 204-208.
  • [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
  • [Wi74] Wilson, Richard M., Concerning the number of mutually orthogonal Latin squares. Discrete Math. (1974), 181-198.

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.