Hypergraph Ramsey numbers: tight cycles versus cliques

mubayi, dhruv; r¨odl, vojtˇech;

For s >= 4, the 3-uniform tight cycle C3 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 >= 4 and
s ≡ 0 (mod 3), we prove that there are positive constants a and b with 2at < r(C3 s,K3 t ) < 2bt2 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(K3 4,K3 t ), while for s = 5 it follows from a new upper bound for r(K3− 5 ,K3t ) that we develop.