On Ramsey choice and partial choice for infinite families ofn-element sets

摘要

For an integer n > 2, Ramsey Choice RC,, is the weak choice principle "every infinite set x has an infinite subset y such that [y]n (the set of all n -element subsets of y) has a choice function", and CW is the weak choice principle "every infinite family of n -element sets has an infinite subfamily with a choice function". In 1995, Montenegro showed that for n = 2, 3, 4, RC C. However, the question of whether or not RC, C,7 for n > 5 is still open. In general, for distinct in, n > 2, not even the status of "RC, > C,77" or "RC > RC," is known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following: