We investigate the induced Ramsey number\(\) of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either 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 \(\), we have
where \(\) is the chromatic number of H and C is some universal constant. Furthermore, we also investigate \(\) imposing some conditions on G. For instance, we prove a bound that is polynomial in both k and t in the case in which G is a tree. Our methods of proof employ certain random graphs based on projective planes.
AMS Subject Classification (1991) Classes: 05C55, 05C80; 05C35
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© János Bolyai Mathematical Society, 1998