Improved Bounds for the Ramsey Number of Tight Cycles Versus Cliques
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The 3-uniform tight cycle Cs3 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(Cs3, Kn3) 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.
1953 ◽
Vol 49
(1)
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pp. 59-62
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1970 ◽
Vol 22
(3)
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pp. 569-581
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2011 ◽
Vol 20
(4)
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pp. 519-527
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2011 ◽
Vol 54
(3)
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pp. 685-693
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