The relationship between some nonclassical Ramsey numbers

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, $\mathrm{\Gamma }(B)\ge m$ or $\mathrm{\Gamma }(R)\ge n$ ($\beta® \geq n$);where$B$and$R$isthesubgraphof$K_p$ induced by blue and red edges, respectively; $\mathrm{\Gamma }(G)$ 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 $K_p$, $IR(B)\ge m$ or $\beta (R)\ge n$ ($IR(G)$ 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.