Problem 161

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

For fixed $n,t$ as we change $\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\alpha)$ increase continuously or are there jumps? Only one jump?

Source: erdosproblems.com/161

Combinatorics
Hypergraphs
Ramsey theory
Discrepancy

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label (page last edited 16 January 2026). The one claim is Conlon, Fox and Sudakov's accepted partial claim [CFS11] (claim page (Conlon Fox Sudakov, 2009)): for $t=3$ the order of growth of $F^{(3)}(n,\alpha)$ is $\sqrt{\log n}$ for every fixed $\alpha\in(0,1/2)$ , so no jump occurs inside $(0,1/2)$ . Whether a jump occurs at $0$ for $t=3$ , and the whole question for $t\ge4$ , are open, so the problem stays open.

CFS10: Conlon, D., Fox, J. and Sudakov, B., Hypergraph Ramsey numbers. J. Amer. Math. Soc. 23 (2010), no. 1, 247--266, DOI 10.1090/S0894-0347-09-00645-6; arXiv:0808.3760v1 (27 August 2008). Section 6.2, pp. 16--17 of the preprint. Library home: Conlon and Sudakov 2008.
CFS11: Conlon, David and Fox, Jacob and Sudakov, Benny, Large almost monochromatic subsets in hypergraphs. Israel J. Math. 181 (2011), no. 1, 423--432, DOI 10.1007/s11856-011-0016-6; arXiv:0901.3912 (25 January 2009). Library home: Conlon, Fox and Sudakov 2011.
Er90b: Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28; pp. 21--22, displays (31)--(32) and the jump question. Library home: Erdős 1990.

Problem 161

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Proof claims

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