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.