Asymptotic Bounds for CO-irredundant and Irredundant Ramsey Numbers

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant (CO-irredundant) if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ ($y\in V$) that is adjacent to $x$ and to no other vertex of $X$. The irredundant Ramsey number $s(t_{1},\ldots,t_{l})$, CO-irredundant Ramsey number $s_{\operatorname{CO}}(t_{1},\ldots,t_{l})$, is the minimum $N$ such that every $l$-coloring of the edges of the complete graph $K_{N}$ on $N$ vertices has a monochromatic irredundant set, a monochromatic CO-irredundant set, of size $t_{i}$ for some $1\leq i\leq l$, respectively. In this paper, firstly, we establish a lower bound for the irredundant Ramsey number $s(t_{1},\ldots,t_{l})$ by a random and probabilistic method. Secondly, we improve an upper bound for $s(3,9)$ such that $24\leq s(3,9)\leq 26$. Thirdly, using Krivelevich's lemma, we establish an asymptotic lower bound for the $\operatorname{CO}$-irredundant Ramsey number $s_{\operatorname{CO}}(m,n)$.