Selective ultrafilters and homogeneity

We develop a game-theoretic approach to partition theorems, like those of Mathias, Taylor, and Louveau, involving ultrafilters. Using this approach, we extend these theorems to contexts involving several ultrafilters. We also develop an analog of Mathias forcing for such contexts and use it to show that the proposition (considered by Laver and Prikry) “every non-trivial c.c.c. forcing adjoins Cohen-generic reals or random reals” implies the non-existence of P-points. We show that, in the model obtained by Lévy collapsing to ω all cardinals below a Mahlo cardinal ;, any countably many selective ultrafilters are mutually generic over the Solovay (Lebesgue measure) submodel. Finally, we show that a certain natural group of self-homeomorphisms of βω-ω, chosen so as to preserve selectivity of ultrafilters, in fact preserves isomorphism types.