Asymptotic Bounds for CO-irredundant and Irredundant Ramsey Numbers

TL;DR

This paper investigates bounds for irredundant and CO-irredundant Ramsey numbers, providing new lower bounds, improving upper bounds for specific cases, and establishing asymptotic bounds using probabilistic and combinatorial methods.

Contribution

It introduces new bounds for irredundant and CO-irredundant Ramsey numbers, including probabilistic lower bounds, an improved upper bound for s(3,9), and asymptotic bounds using Krivelevich's lemma.

Findings

Established a lower bound for s(t_1,...,t_l) via probabilistic methods.

Improved the upper bound for s(3,9) to between 24 and 26.

Derived an asymptotic lower bound for s_CO(m,n) using Krivelevich's lemma.

Abstract

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 $s_{\operatorname{CO}}(m,n)$.