# The relationship between some nonclassical Ramsey numbers

## TL;DR

This paper investigates nonclassical Ramsey numbers related to domination and irredundant sets, proving key equalities, determining two unknown numbers, and solving four open cases in the field.

## Contribution

It establishes the equality between two nonclassical Ramsey numbers, determines two previously unknown numbers, and resolves four open problems from prior research.

## Key findings

- Proved that v(3,n)=t(3,n) for certain Ramsey numbers.
- Determined v(3,7)=18 and v(3,8)=22.
- Solved four open cases from previous studies.

## Abstract

The upper (mixed) domination Ramsey number $u(m, n)$($v(m,n)$) is the smallest integer $p$ such that every $2$-coloring of the edges of $K_p$ with color red and blue, $\Gamma(B) \geq m$ or $\Gamma(R) \geq n$ ($\beta(R) \geq n$); where $B$ and $R$ is the subgraph of $K_p$ induced by blue and red edges, respectively; $\Gamma(G)$ is the maximum cardinality of a minimal dominating set of a graph $G$. First, we prove that $v(3,n)=t(3,n)$ where $t(m,n)$ is the mixed irredundant Ramsey number i.e. the smallest integer $p$ such that in every two-coloring $(R, B)$ of the edges of $K_p$, $IR(B) \geq m$ or $\beta(R) \geq n$ ($IR(G)$ is the maximum cardinality of an irredundant set of $G$). To achieve this result we use a characterization of the upper domination perfect graphs in terms of forbidden induced subgraphs. By the equality we determine two previously unknown Ramsey numbers, namely $v(3,7)=18$ and $v(3,8) = 22$. In addition, we solve other four remaining open cases from Burger's {\it et. al.} article, which listed all nonclassical Ramsey numbers. We find that $u(3,7)=w(7,3)=18$, $u(3,8) = w(8,3) = 21$, where $w(m,n)$ is the irredundant-domination Ramsey number introduced by Burger and Van Vuuren in 2011.

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Source: https://tomesphere.com/paper/1701.08674