t(3,n)的渐近界及其在t(4,n)中的应用
Asymptotic Bounds for t(3,n) and an Application to t(4,n)
AI总结：
本文确定了t(3,n)和s(3,n)的渐近界，改进了Rousseau和Speed的上界，并验证了Chen等人关于t(4,n)的猜想。
摘要：
在简单图G(V,E)中，顶点集X⊆V若每个顶点x∈X在诱导子图G[X]中孤立或有唯一的邻居y∈V\X，则称X为不可冗余集。混合Ramsey数t(m,n)是使每种红蓝边着色的K_N中存在m个不可冗余集或n个独立集的最小N。不可冗余Ramsey数s(m,n)是使每种红蓝边着色的K_N中存在m个不可冗余集或n个不可冗余集的最小N。本文通过证明t(3,n)=O(n^{5/4}/logn)确定了t(3,n)和s(3,n)的渐近界，改进了Rousseau和Speed的上界。作为应用，本文验证了Chen、Hattingh和Rousseau提出的m=4时的猜想。
英文摘要：
A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other vertex of $X$. The \emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in the blue subgraph or an $n$-element independent set in the red subgraph. The irredundant Ramsey number $s(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in the blue subgraph or an $n$-element irredundant set in the blue subgraph. In this paper, we determine $t(3,n)$ and $s(3,n)$ up to a constant factor by showing that $t(3,n)=O\left(n^{5/4}/{\log{n}}\right)$, which improved the best upper bound due to Rousseau and Speed in [Comb. Probab. Comput. 12 (2003), 653-660]. As an application, we verify a conjecture for $m=4$ proposed by Chen, Hattingh, and Rousseau in [J. Graph Theory 17(2) (1993), 193-206].
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。
