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For an undirected simple graph G, we write $$G \rightarrow (H_1, H_2)^v$$ G→(H1,H2)v if and only if for everyred-blue coloring of its vertices there exists a red $$H_1$$ H1 or a blue $$H_2$$ H2 . Thegeneralized vertex Folkman number $$F_v(H_1, H_2; H)$$ Fv(H1,H2;H) is defined as the smallest integer n for which there exists an H-free graph G of order n such that $$G \rightarrow (H_1, H_2)^v$$ G→(H1,H2)v...
Edge-colorings of multigraphs are studied where a generalization of Ramsey numbers is given. Let $${M_n^{(r)}}$$ be the multigraph of order n, in which there are r edges between any two different vertices. Suppose q1, q2, . . . , qk and r are positive integers, and qi ≥ 2(1 ≤ i ≤ k), k > r. Let the multigraph Ramsey number $${f^{(r)} (q_1 ,q_2 , \ldots ,q_k )}$$ be the minimum...
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