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Erdős #161 · combinatorics / hypergraphs

Let α∈[0,1/2)\alpha\in[0,1/2) and n,t≥1n,t\geq 1. Let F(t)(n,α)F^{(t)}(n,\alpha) be the smallest mm such that we can 22-colour the edges of the complete tt-uniform hypergraph on nn vertices such that if X⊆[n]X\subseteq [n] with $\lvert …

MSC 05 · Combinatorics

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Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let α∈[0,1/2)\alpha\in[0,1/2) and n,t≥1n,t\geq 1. Let F(t)(n,α)F^{(t)}(n,\alpha) be the smallest mm such that we can 22-colour the edges of the complete tt-uniform hypergraph on nn vertices such that if X⊆[n]X\subseteq [n] with ∣X∣≥m\lvert X\rvert \geq m then there are at least α(∣X∣t)\alpha \binom{\lvert X\rvert}{t} many tt-subsets of XX of each colour.

For fixed n,tn,t as we change α\alpha from 00 to 1/21/2 does F(t)(n,α)F^{(t)}(n,\alpha) increase continuously or are there jumps? Only one jump?

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Background

For α=0\alpha=0 this is the usual Ramsey function.

A conjecture of Erdős, Hajnal, and Rado (see [562]) implies that

F(t)(n,0)≍log⁡t−1nF^{(t)}(n,0)\asymp \log_{t-1} n

and results of Erdős and Spencer imply that

F(t)(n,α)≫α(log⁡n)1t−1F^{(t)}(n,\alpha) \gg_\alpha (\log n)^{\frac{1}{t-1}}

for all α>0\alpha>0, and a similar upper bound holds for α\alpha close to 1/21/2.

Erdős said in [Er90b]: 'If I can hazard a guess completely unsupported by evidence, I am afraid that the jump occurs all in one step at 00. It would be much more interesting if my conjecture would be wrong and perhaps there is some hope for this for t>3t>3. I know nothing and offer $500 to anybody who can clear up this mystery.'

Conlon, Fox, and Sudakov [CFS11] have proved that, for any fixed α>0\alpha>0,

F(3)(n,α)≪αlog⁡n.F^{(3)}(n,\alpha) \ll_\alpha \sqrt{\log n}.

Coupled with the lower bound above, this implies that there is only one jump for fixed α\alpha when t=3t=3, at α=0\alpha=0.

For all α>0\alpha>0 it is known that

F(t)(n,α)≫t(log⁡n)cα.F^{(t)}(n,\alpha)\gg_t (\log n)^{c_\alpha}.

See also [563] for more on the case t=2t=2.

This problem is #40 in Ramsey Theory in the graphs problem collection.

Progress & source assessment

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 #161 link

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