3 research outputs found
On the Nonexistence of Some Generalized Folkman Numbers
For an undirected simple graph , we write if and only if for every red-blue coloring of its vertices there exists a red or a blue . The generalized vertex Folkman number is defined as the smallest integer for which there exists an -free graph of order such that . The generalized edge Folkman numbers are defined similarly, when colorings of the edges are considered. We show that and are well defined for . We prove the nonexistence of for some , in particular for , where is the book graph of triangular pages, and for . We pose three problems on generalized Folkman numbers, including the existence question of edge Folkman numbers , and . Our results lead to some general inequalities involving two-color and multicolor Folkman numbers