Star-Avoiding Ramsey Numbers

The graph Ramsey number R(G,H) is the smallest integer n such that every 2-coloring of the edges of Kn contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kn such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-avoiding Ramsey number r∗(G,H) as the smallest integer k such that every 2-coloring of the edges of Kn −K1,n−1−k contains either a red copy of G or a blue copy of H . We find the star-avoiding Ramsey number for trees versus complete graphs, multiple copies of K2 and K3, and paths versus a 4-cycle. In addition to finding the star-avoiding Ramsey numbers, the critical graphs are classified for R(Tn,Km), R(nK2,mK2) and R(Pn, C4).