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Erdős #1183 · combinatorics / ramsey theory

Let f(n)f(n) be maximal such that in any 22-colouring of the subsets of {1,…,n}\{1,\ldots,n\} there is always a monochromatic family of at least f(n)f(n) sets which is closed under taking unions and intersections. Estimate $f(n)…

MSC 05 · Combinatorics

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

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Problem statement

Let f(n)f(n) be maximal such that in any 22-colouring of the subsets of {1,…,n}\{1,\ldots,n\} there is always a monochromatic family of at least f(n)f(n) sets which is closed under taking unions and intersections. Estimate f(n)f(n).

Let F(n)F(n) be defined similarly, except that we only require the family be closed under taking unions. Estimate F(n)F(n). In particular, is it true that F(n)≥nω(n)F(n)\geq n^{\omega(n)} for some ω(n)→∞\omega(n)\to \infty as n→∞n\to \infty, and F(n)<(1+o(1))nF(n)<(1+o(1))^n?

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Background

A problem of Erdős and Ulam.

It is trivial that f(n)≥n+12f(n)\geq \frac{n+1}{2}, since there is a sequence of n+1n+1 many nested subsets. This seems to be all that they knew; Erdős [Er78] wrote 'we have no plausible conjecture for the true order of magnitude of f(n)f(n)'.

They similarly had no good guess about F(n)F(n).

In [Er78] Erdős wrote that if the colouring is such that all subsets of same size receive the same colour then Howorka had proved that F(n)>nω(n)F(n)>n^{\omega(n)} for some ω(n)→∞\omega(n)\to \infty, but gave no reference.

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The source labels this entry OPEN 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 #1183 link

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