On Ramsey numbers of complete graphs with dropped stars

Let r(G,H)r(G,H) be the smallest integer NN such that for any 22-coloring (say, red and blue) of the edges of KnKn, n⩾Nn⩾N, there is either a red copy of GG or a blue copy of HH. Let Kn−K1,sKn−K1,s be the complete graph on nn vertices from which the edges of K1,sK1,s are dropped. In this note we present exact values for r(Km−K1,1,Kn−K1,s)r(Km−K1,1,Kn−K1,s) and new upper bounds for r(Km,Kn−K1,s)r(Km,Kn−K1,s) in numerous cases. We also present some results for the Ramsey number of Wheels versus Kn−K1,sKn−K1,s.