The nonclassical mixed domination Ramsey numbers

The nonclassical mixed domination Ramsey number v(m,G) is the smallest integer p such that in every 2-coloring of the edges of Kp with color red and blue, either Γ(B) ≥ m or there exists a blue copy of graph G, where B is the subgraph of Kp induced by blue edges. Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. We give exact values for numbers v(m,K3 − e), v(3, Pm), v(3, Cm). In addition, we give exact values and bounds for numbers v(3, Kn − e), where n ≥ 3.