Improved Bounds for the Ramsey Number of Tight Cycles Versus Cliques

The 3-uniform tight cycle C s has vertex set Zs and edge set {{i, i+ 1, i+ 2} : i ∈ Zs}. We prove that for every s 6≡ 0 (mod 3) and s ≥ 16 or s ∈ {8, 11, 14} there is a cs > 0 such that the 3-uniform hypergraph Ramsey number r(C s ,K n ) satisfies r(C s ,K n ) < 2cn . This answers in strong form a question of the author and Rödl who asked for an upper bound of the form 2n 1+ǫs for each fixed s ≥ 4, where ǫs → 0 as s → ∞ and n is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in n.