The First Nontrivial Three Color Upper Domination Ramsey Number is 13

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 \(\Gamma(\overline{G}) \geq 3\), where \(\Gamma(\overline{G})\) 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 \(13 \leq U(3, 3, 3) \leq 14\) provided by Michael A. Henning and Ortrud R. Oellermann.