Bounds for some generalized vertex Folkman numbers

For a graph \(G\) and positive integers \(a_1,…,a_r,\) if every r-coloring of vertics V(G) must result in a monochromatic \(a_1\)-clique of color \(i\) for some \(i \in \{1,…,r\},\) then we write \(G \to (a_1,..a_r)^v\).\(F_v(K_a1,…,K_ar;H)\) is the smallest integer \(n\) such that there is an H-free graph \(G\) of order \(n\), and \(G \to (a_1,…,a_r)^v\). In this paper we study upper and lower bounds for some generalized vertex Folkman numbers of from \(F_v(K_{a1},…,K_{ar};K_4 – e)\), where \(r \in {2,3}\) and \(a_1 \in {2,3}\) for 10 and \(F_v(K_2,K_3;K_4 – e) = 19\) by computing, and prove \(F_v(K_3,K_3;K_4 – e)\ge F_v(K_2,K_2,K_3;K_4 – e)\ge 25\)