The Ramsey number of loose cycles versus cliques

TL;DR

This paper establishes new upper bounds on the Ramsey numbers of loose cycles versus cliques in hypergraphs, improving previous results and confirming conjectures for specific cycle lengths and uniformities.

Contribution

It provides the first non-trivial upper bounds for the Ramsey numbers of loose cycles of length five versus cliques in 3-uniform hypergraphs and extends bounds to other cycle lengths and uniformities.

Findings

Proves that R(C_5^3, K_n^3) = O(n^{4/3})

Generalizes bounds to R(C_l^3, K_n^3) = O(n^{1 + 1/⌊(l+1)/2⌋}) for l ≥ 3

Improves bounds for fixed l ≥ 5, r ≥ 4, showing R(C_l^r, K_n^r) = O(n^{1 + 1/⌊l/2⌋})

Abstract

Recently Kostochka, Mubayi and Verstra\"ete initiated the study of the Ramsey numbers of uniform loose cycles versus cliques. In particular they proved that $R(C^r_3,K^r_n) = \tilde{\theta}(n^{3/2})$ for all fixed $r\geq 3$. For the case of loose cycles of length five they proved that $R(C_5^r,K_n^r)=\Omega((n/\log n)^{5/4})$ and conjectured that $R(C^r_5,K_n^r) = O(n^{5/4})$ for all fixed $r\geq 3$. Our main result is that $R(C_5^3,K_n^3) = O(n^{4/3})$ and more generally for any fixed $l\geq 3$ that $R(C_l^3,K_n^3) = O(n^{1 + 1/\lfloor(l+1)/2 \rfloor})$. We also explain why for every fixed $l\geq 5$, $r\geq 4$, $R(C^r_l,K^r_n) = O(n^{1+1/\lfloor l/2 \rfloor})$ if $l$ is odd, which improves upon the result of Collier-Cartaino, Graber and Jiang who proved that for every fixed $r\geq 3$, $l\geq 4$, we have $R(C_l^r,K_n^r) = O(n^{1 + 1/(\lfloor l/2 \rfloor-1)})$.