On $K_{4}^{\prime}$-graphs

Abstract

$G$ is called $K_4'$-graph if for every 2-coloring of its edges there are monochromatic triangles with common edge. By $R^k(K_4')$ we denote the minimum of vertex number of $K_4'$-graphs with clique number $k$. The inequalities $R^5(K_4')\leq 29$ and $R^4(K_4')\leq 61$ are proved.