Induced Ramsey Numbers

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.

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Received: October 10, 1997

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Kohayakawa,, Y., Prömel, H. & Rödl, V. Induced Ramsey Numbers. Combinatorica 18, 373–404 (1998). https://doi.org/10.1007/PL00009828

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  • AMS Subject Classification (1991) Classes:  05C55, 05C80; 05C35