Induced Ramsey numbers

We investigate the induced Ramsey number r(ind)(G,H) of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Gam ma with the property that, whenever its edges are coloured red and blue, ei ther a red induced copy of G arises or else a blue induced copy of H arises . We show that, for any G and H with k = \V(G)\less than or equal to t=\V(H )\, we have r(ind)(G, H)less than or equal to t(Ck log q), where q = chi(H) is the chromatic number of H and C is some universal const ant. Furthermore, we also investigate r(ind)(G,H) imposing some conditions on G. For instance, we prove abound that is polynomial in both k and t in t he case in which G is a tree. Our methods of proof employ certain random gr aphs based on projective planes.