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Level A · Machine-checkable Hard Combinatorics P-erdos-725

Erdős Problem #725

Give an asymptotic formula for the number of k× n Latin rectangles. The formula F(k,n) must satisfy L(k,n)∼ F(k,n) as n→∞ for every choice of k=k(n)≤ n, so it also covers Latin squares (k=n).

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-725,
  title        = {Erdős Problem #725},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-725}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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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

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

The problem

The question

Give an asymptotic formula for the number of Latin rectangles.

The formula must satisfy as for every choice of , so it also covers Latin squares ().

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«725».

theorem erdos_725 (k : ℕ → ℕ) (hk : ∀ n, k n ≤ n) :
    (fun n ↦ (L (k n) n : ℝ)) ~[atTop] fun n ↦ erdos_725.answer (k n) 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/725. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/725
  • [Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica 1 (1981), 25-42. PDF
  • [ErKa46] Erdős, P. and Kaplansky, I., The asymptotic number of Latin rectangles. Amer. J. Math. 68 (1946), 230-236. PDF
  • [Ya51] Yamamoto, K., On the asymptotic number of Latin rectangles. Jpn. J. Math. 21 (1951), 113-119.

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.