On the Induced Ramsey Number IR(P3, H).

The induced Ramsey number IR(G, H) is the least positive integer N such that there exists an N-vertex graph F with the property that for each 2-coloring of its edges with red and blue, either some induced in F subgraph isomorphic to G has all its edges colored red, or some induced in F subgraph isomorphic to H has all its edges colored blue. In this paper, we study IR(P3, H) for various H, where P3 is the path with 3 vertices. In particular, we answer a question by Gorgol and Luczak by constructing a family {Hnn=1∞ such that $$ \mathop {lim}\limits_{n \to \infty } \sup \tfrac{{IR(P_3 ,H_n )}} {{IR_w (P_3 ,H_n )}} > 1 $$, where IRw(G, H) is defined almost as IR(G,H), with the only difference that G should be induced only in the red subgraph of F (not in F itself) and H should be induced only in the blue subgraph of F. [ABSTRACT FROM AUTHOR]