SOME COMBINATORIAL PROPERTIES OF SEMISELECTIVE IDEALS

Abstract

Abstract We present several combinatorial properties of semiselective ideals on the set of natural numbers. The continuum hypothesis implies that the complement of every selective ideal contains a selective ultrafilter, however for semiselective ideals this is not the case. We prove that under certain hypothesis, for instance, $V=L$ , there are semiselective ideals whose complement does not contain a selective ultrafilter, and that it is also consistent that the complement of every semiselective ideal contains a selective ultrafilter; specifically, we show that if $V=L$ then there is a generic extension of V where this occurs. We present other results concerning semiselective ideals, namely, an alternative proof of Ellentuck’s theorem for the local Ramsey property, and we prove some facts about the additivity of the ideal of local Ramsey null sets, and also about the generalized Suslin operation on the algebra of local Ramsey sets.