On Some Generalized Vertex Folkman Numbers

For a graph G and integers $$a_i\ge 1$$ , the expression $$G \rightarrow (a_1,\ldots ,a_r)^v$$ means that for any r-coloring of vertices there exists monochromatic $$a_i$$ -clique in some color $$i \in \{1,\ldots ,r\}$$ . The vertex Folkman numbers are defined as $$F_v(a_1,\ldots ,a_r;H) = \min \{|V(G)| : G$$ is H-free ,a_r)^v\}$$ where H graph. Such have been extensively studied $$H=K_s$$ with $$s>\max \{a_i\}_{1\le i \le r}$$ If $$a_i=a$$ all i, then we use notation $$F_v(a^r;H)=F_v(a_1,\ldots ,a_r;H)$$ Let $$J_k$$ be complete $$K_k$$ missing one edge, i.e. $$J_k=K_k-e$$ In this work focus on $$H=J_k$$ particular $$k=4$$ $$a_i\le 3$$ A result by Nešetřil Rödl from 1976 implies $$F_v(3^r;J_4)$$ well $$r\ge 2$$ We present new more direct proof fact. simplest but already intriguing case $$F_v(3,3;J_4)$$ which establish upper bound 135 using $$J_4$$ -free process. obtain exact values bounds few other small cases ,a_r;J_4)$$ when $$a_i $$1 r$$ including $$F_v(2,3;J_4)=14$$ $$F_v(2^4;J_4)=15$$ $$22 F_v(2^5;J_4) 25$$ Note $$F_v(2^r;J_4)$$ smallest number chromatic $$r+1$$ Most results were obtained help computations, graphs found interesting themselves.