Hypergraph Ramsey numbers: tight cycles versus cliques

TL;DR

This paper establishes exponential bounds on the 3-uniform hypergraph Ramsey numbers for tight cycles versus cliques, using probabilistic methods and supersaturation techniques, with new bounds for specific cases.

Contribution

It provides the first bounds for hypergraph Ramsey numbers involving tight cycles and introduces new upper bounds for specific cases like $s=5$.

Findings

Lower bounds via probabilistic construction

Upper bounds using supersaturation and known results

New bounds for $r(K_5^{3-}, K_t^3)$

Abstract

For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at}<r(C^3_s, K^3_t)<2^{bt^2\log t}.$$ The lower bound is obtained via a probabilistic construction. The upper bound for $s>5$ is proved by using supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for $s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we develop.