LOWER BOUNDING THE FOLKMAN NUMBERS F_v(a_1, . . . , a_s;m − 1) - Bikov, Aleksandar; Nenov, Nedyalko (Ann. Sofia Univ. Fac. Math. and Inf., 2017)
For a graph G the expression Gv→(a_1,...,a_s) means that for every s-coloring of the vertices of G there exists i{1,2,...,s} there exists a monochromatic a_i-clique of color i. The vertex Folkman numbers \[ F_v(a_1,...,a_s;m-1) ={|V(G)|:Gv→(a_1,...,a_s) and K_m-1 subseteq G}\] are considered, where m = _i=1^s(a_i - 1) + 1. We know the exact values of all the numbers F_v(a_1, . . . , a_s;m − 1) when max{a_1, . . . , a_s} ≤ 6 and also the number F_v(2, 2, 7; 8) = 20. In [1] we present a method for obtaining lower bounds on these numbers. With the help of this method and a new improved algorithm, in the special case when max{a_1, . . . , a_s} = 7 we prove that F_v(a_1, . . . , a_s;m − 1) ≥ m + 11 and this bound is exact for all m. The known upper bound for these numbers is m + 12. At the end of the paper we also prove the lower bounds 19 ≤ F_v(2, 2, 2, 4; 5) and 29 ≤ F_v(7, 7; 8).