The Ramsey number of loose cycles versus cliques

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 $R(C_l^r,K_n^r) = O(n^{1 + 1/(\lfloor l/2 \rfloor-1)})$.