The First Nontrivial Three Color Upper Domination Ramsey Number is 13

Abstract

The upper domination Ramsey number $u(3,3,3)$ is the smallest integer $n$ such that every $3$-coloring of the edges of complete graph $K_n$ contains a monochromatic graph $G$ with $\mathrm{\Gamma }(\overline{G})\ge 3$, where $\mathrm{\Gamma }(\overline{G})$ is the maximum order over all the minimal dominating sets of the complement of $G$. In this note, with the help of computers, we determine that $U(3,3,3)=13$, which improves the results that $13\le U(3,3,3)\le 14$ provided by Michael A. Henning and Ortrud R. Oellermann.