On some edge Folkman numbers, small and large

Abstract

Edge Folkman numbers $F_e\left(G_1,G_2;k\right)$ can be viewed as a generalization of more commonly studied Ramsey numbers. $F_e\left(G_1,G_2;k\right)$ is defined as the smallest order of any $K_k$-free graph $F$ such that any red-blue coloring of the edges of $F$ contains either a red $G_1$ or a blue $G_2$. In this note, first we discuss edge Folkman numbers involving graphs $J_s=K_s-e$, including the results $F_e\left(J_3,K_n;n+1\right)=2n-1$, $F_e\left(J_3,J_n;n\right)=2n-1$, and $F_e\left(J_3,J_n;n+1\right)=2n-3$. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on $F_e\left(J_4,J_4;k\right)$ for all $k>4$.