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OP-9208OpenGraph theory

Erdős #835 · graph theory / hypergraphs

Does there exist a k>2k>2 such that the kk-sized subsets of {1,…,2k}\{1,\ldots,2k\} can be coloured with k+1k+1 colours such that for every A⊂{1,…,2k}A\subset \{1,\ldots,2k\} with ∣A∣=k+1\lvert A\rvert=k+1 all k+1k+1 colours appear among the…

MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Does there exist a k>2k>2 such that the kk-sized subsets of {1,…,2k}\{1,\ldots,2k\} can be coloured with k+1k+1 colours such that for every A⊂{1,…,2k}A\subset \{1,\ldots,2k\} with ∣A∣=k+1\lvert A\rvert=k+1 all k+1k+1 colours appear among the kk-sized subsets of AA?

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Formal statement

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Background

A problem of Erdős and Rosenfeld. This is trivially possible for k=2k=2. They were not sure about k=6k=6.

This is equivalent to asking whether there exists k>2k>2 such that the chromatic number of the Johnson graph J(2k,k)J(2k,k) is k+1k+1 (it is always at least k+1k+1 and at most 2k2k). The chromatic numbers listed at this website show that this is false for 3≤k≤83\leq k\leq 8.

Ma and Tang have proved that the chromatic number of J(2k,k)J(2k,k) is >k+1>k+1 for all k>2k>2 not of the form p−1p-1 for prime pp.

Progress & source assessment

The source labels this entry VERIFIABLE in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #835 link
  2. [2]Upstream Lean 4 statement for Erdős #835 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 835 link

Recorded progress

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