Erdős (Erdos) Problem #901: status in 2026

Erdős Problem #901

Let m(n) be minimal such that there is an n-uniform hypergraph with m(n) edges which is 3-chromatic. Estimate m(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-901,
  title        = {Erdős Problem #901},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-901}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Disputed

Refuted

On the literature board

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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.

The problem

The question

Let $m (n)$ be minimal such that there is an $n$ -uniform hypergraph with $m (n)$ edges which is $3$ -chromatic. Estimate $m (n)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«901».

theorem erdos_901 :
    (fun n ↦ ((Hypergraph.minChromaticEdges n 3).toNat : ℝ)) =Θ[atTop]
    erdos_901.answer

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/901. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/901
[Be77] Beck, J., On a combinatorial problem of {P}. {E}rdős and {L}. {L}ovász. Discrete Math. (1977), 127--131.
[Be78] Beck, J., On { $3$ }-chromatic hypergraphs. Discrete Math. (1978), 127--137.
[Er63b] Erdős, P., On a combinatorial problem. Nordisk Mat. Tidskr. (1963), 5--10, 40.
[Er64e] Erdős, P., On a combinatorial problem. {II}. Acta Math. Acad. Sci. Hungar. (1964), 445--447.
[ErLo75] Erdős, P. and Lovász, L., Problems and results on { $3$ }-chromatic hypergraphs and some related questions. (1975), 609--627.
[Pl09] Pluhár, András, Greedy colorings of uniform hypergraphs. Random Structures Algorithms (2009), 216--221.
[RaSr00] Radhakrishnan, Jaikumar and Srinivasan, Aravind, Improved bounds and algorithms for hypergraph { $2$ }-coloring. Random Structures Algorithms (2000), 4--32.

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.