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arXiv 1603.01697math.CO
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$4$ -均匀松弛圈的Ramsey数

Ramsey numbers of $4$-uniform loose cycles

Gholamreza Omidi, Maryam Shahsiah

备注：
arXiv admin note: substantial text overlap with arXiv:1602.05386
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摘要：

Gyárfás，Sárközy和Szemerédi证明了一个$k$ -均匀松弛圈$\mathcal{C}^k_n$的$2$ -色Ramsey数$R(\mathcal{C}^k_n,\mathcal{C}^k_n)$是渐近的$\frac{1}{2}(2k-1)n,$，它产生了与Haxell等人关于$k=3$的结果相同的结果。最近，作者证明了这个猜想对$n=m\geq 2$和$k\geq 8$是正确的。本文证明了当$n>m$或$n=m$为奇数时，该猜想对$k=4$成立。当$n=m$为偶数时，我们证明了$R(\mathcal{C}^4_n,\mathcal{C}^4_n)$介于两个值之间，差为1。

英文摘要：

Gyárfás, Sárközy and Szemerédi proved that the $2$-color Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ of a $k$-uniform loose cycle $\mathcal{C}^k_n$ is asymptotically $\frac{1}{2}(2k-1)n,$ generating the same result for $k=3$ due to Haxell et al. Concerning their results, it is conjectured that for every $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_m)=(k-1)n+\lfloor\frac{m-1}{2}\rfloor.$$ In $2014$, the case $k=3$ is proved by the authors. Recently, the authors showed that this conjecture is true for $n=m\geq 2$ and $k\geq 8$. In this paper, we show that the conjecture holds for $k=4$ when $n>m$ or $n=m$ is odd. When $n=m$ is even, we show that $R(\mathcal{C}^4_n,\mathcal{C}^4_n)$ is between two values with difference one.

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