Improved bounds for the Ramsey number of tight cycles versus cliques

Abstract

The 3-uniform tight cycle $C_s^3$ has vertex set $Z_s$ and edge set $\{\{i,i+1,i+2\}:i\in Z_s\}$. We prove that for every $s\not\equiv 0$ (mod 3) and $s\ge 16$ or $s\in \{8,11,14\}$ there is a $c_s>0$ such that the 3-uniform hypergraph Ramsey number $$r(C_s^3,K_n^3)<2^{c_{s}n\log n}$$ This answers in strong form a question of the author and Rödl who asked for an upper bound of the form $2^{n^{1+{\epsilon }_s}}$ for each fixed $s\ge 4$, where ${\epsilon }_s\to 0$ as $s\to \infty$ and $n$ is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in $n$.

References in corpus (1)

• Hypergraph Ramsey numbers: tight cycles versus cliques