The upper domination Ramsey number \(u(3,3,3)\) (Q5957750)

The upper domination number of a graph is the maximal number of vertices in a minimal dominating set. The upper domination Ramsey number \(u(n_1, n_2, \dots, n_k)\) is the smallest integer \(n\) such that for any \(k\)-coloring of the edges of a \(K_n\), the \(i\)th color graph will have upper domination number at least \(n_i\) for some \(i\). It is shown that \(u(3,3,3) = 13\) or \(14\), which verifies that the upper domination Ramsey number \(u(3,3,3)\) is less than the Ramsey number \(r(3,3,3) = 17\), and could possibly equal the irredundant Ramsey number \(s(3,3,3) = 13\).