Irredundant sets, Ramsey numbers, multicolor Ramsey numbers

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 a blue subgraph or a $n$-element independent set in a red subgraph. The \emph{multicolor irredundant Ramsey number} $s(t_{1},\ldots,t_{l})$ is the minimum $r$ such that every $l$-coloring of the edges of the complete graph $K_{r}$ on $r$ vertices has a monochromatic irredundant set of size $s_{i}$ for certain $1\leq i\leq l$. Firstly, we improve the upper bound for the mixed Ramsey number $t(3,n)$, and using this result, we verify a special case of a conjecture proposed by Chen, Hattingh, and Rousseau for $m=4$. Secondly, we obtain a new upper bound for $s(3,9)$, and using Krivelevich's method, we establish an asymptotic lower bound for CO-irredundant Ramsey number of $K_{N}$, which extends Krivelevich's result on $s(m,n)$. Thirdly, we prove a lower bound for the multicolor irredundant Ramsey number by a random and probability method which has been used to improve the lower bound for multicolor Ramsey numbers. Finally, we give a lower bound for the irredundant multiplicity.Comment: 23 pages, 1 figur