Intersection properties of functions on cardinals

Abstract

This thesis consists of three parts, each one concerned with a problem in Combinatorial Set Theory.Part 1 deals with problems involving set mappings of unrestricted order. It is well-known that set mappings of order less than K always have a free set of size K . We prove an inversion theorem which allows us to apply existing results about set mappings of order K to get results about set mappings of unrestricted order. See [3]. Part 2 considers generalizations of the regressive functions studied by Fodor in his classical paper [2]. Fodor showed that if a function f : K -*• K satisfies /(a) < a on a set that is stationary in K , then / is constant on a stationary subset of this set. We develop conditions that guarantee that there exists a stationary constant set for a function / : K -*• [ K ]~ which satisfies /(a) c a on some subset 5 of < . We K.VC also consider the stationary subsets of the set [p]Part 3 is the main part of the thesis. We study two generalizations of property B . A family of sets A has property B if there is a set T such that A n T + 0 and yet A <£ T for all sets A in A . These two requirements may be restated in terms of the characteristic functions of the sets in A as follows: 3x(Xy(x) = XA X ) ~ l) a n d