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arXiv 1909.01461math.COcs.DM
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关于伪随机Ramsey图的一个注记

A note on pseudorandom Ramsey graphs

Dhruv Mubayi, Jacques Verstraete

备注：
10 pages, minor changes to the first version
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摘要：

对于固定的$s \ge 3$，证明了如果存在最优的无$K_s$的伪随机图，则Ramsey数$r(s,t) = t^{s-1+o(1)}$为$t \rightarrow \infty$ .我们的方法还改进了由Bohman和Keevash从随机$C_{\ell}$ -free过程中获得的$r(C_{\ell},t)$的最佳下界，对所有奇数$\ell \geq 5$和$\ell \in \{6,10\}$都使用了多对数因子。对于$\ell = 4$，它与无$C_4$过程的下限相匹配.我们还证明，通过不同的方法，$r(C_5, t)> (1+o(1))t^{11/8}$和$r(C_7, t)> (1+o(1))t^{11/9}$。这些改善指数的$t$在以前的最佳结果，似乎是第一个例子的图形$F$与周期，这种改善的指数$r(F, t)$显示在边界上给出的随机$F$ -免费的过程和随机图。

英文摘要：

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by Bohman and Keevash from the random $C_{\ell}$-free process by polylogarithmic factors for all odd $\ell \geq 5$ and $\ell \in \{6,10\}$. For $\ell = 4$ it matches their lower bound from the $C_4$-free process. We also prove, via a different approach, that $r(C_5, t)> (1+o(1))t^{11/8}$ and $r(C_7, t)> (1+o(1))t^{11/9}$. These improve the exponent of $t$ in the previous best results and appear to be the first examples of graphs $F$ with cycles for which such an improvement of the exponent for $r(F, t)$ is shown over the bounds given by the random $F$-free process and random graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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