Erdős Problem #835 — open problem in combinatorics

Erdős Problem #835

Does there exist a k>2 such that the k-sized subsets of 1,...,2k can be coloured with k+1 colours such that for every A⊂ 1,…,2k with lvert Arvert=k+1 all k+1 colours appear among the k-sized subsets of A?

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-835,
  title        = {Erdős Problem #835},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-835}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Does there exist a $k > 2$ such that the $k$ -sized subsets of {1,...,2k} can be coloured with $k + 1$ colours such that for every $A \subset {1, \dots, 2 k}$ with $∣ A ∣ = k + 1$ all $k + 1$ colours appear among the $k$ -sized subsets of $A$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«835». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_835 : (∃ k > 2, Property k) ↔ answer(sorry)

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/835. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_835.variants.johnson — Alternative statement of Erdős Problem 835 using the chromatic number of the Johnson graph.

References

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.