The Ramsey number of loose cycles versus cliques

Let R(C-l(r), K-n(r)) be the Ramsey number of an r-uniform loose cycle of length l versus an r-uniform clique of order n. Kostochka et al. showed that for each fixed r >= 3, the order of magnitude of R(C-3(r), K-n(r)) is n(3/2) up to a polylogarithmic factor in n. They conjectured that for each r >= 3 we have R(C-5(r), K-n(r)) = O(n(5/4)). We prove that R(C-5(3), K-n(3)) = O(n(4/3)), and more generally for every l >= 3 that R(C-l(3), K-n(3)) = O(n(1+1/left perpendiclaur (1+1)/2 right perpendicular)). We also prove that for every l >= 5 and r >= 4, R(C-l(r), K-n(r)) = O(n(1+1/left perpendiclaur) l/2 (right perpendiclaur)) if l is odd, which improves upon the result of Collier-Cartaino et al. who proved that for every r >= 3 and l >= 4 we have R(C-l(r), K-n(r)) = O(n1+1/((left perpendiclaur) l/2 (right perpendiclaur) -1).