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Improved Bounds for the Ramsey Number of Tight Cycles Versus Cliques



The 3-uniform tight cycle Cs 3 has vertex set ${\mathbb Z}_s$ and edge set {{i, i + 1, i + 2}: i ${\mathbb Z}_s$ }. We prove that for every s ≢ 0 (mod 3) with s ⩾ 16 or s ∈ {8, 11, 14} there is a cs > 0 such that the 3-uniform hypergraph Ramsey number r(Cs 3, Kn 3) satisfies

$$\begin{equation*} r(C_s^3, K_n^3)< 2^{c_s n \log n}.\ \end{equation*}$$
This answers in a strong form a question of the author and Rödl, who asked for an upper bound of the form $2^{n^{1+\epsilon_s}}$ for each fixed s ⩾ 4, where εs → 0 as s → ∞ and n is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in n.



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Improved Bounds for the Ramsey Number of Tight Cycles Versus Cliques



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