abstract
On Ramsey Choice and Partial Choice for infinite families of n-element sets
Lorenz Halbeisen and Eleftherios Tachtsis
For an integer n, let Ramsey Choice RC(n) be the
weak choice principle "every infinite set X has an infinite subset Y
such that the set of all n-element subsets of Y has a choice
function", and let C(n) be the weak choice principle "every
infinite family of n-element sets has a partial choice function".
In 1995, Montenegro has shown that for n=2,3,4, RC(n) implies C(n).
However, the question of whether RC(n) implies C(n) for
n bigger than 4 is still open. In general, for distinct positive integers
m,n, not even the status of "RC(m) implies C(n)" or "RC(m) implies RC(n)" is known.
In this paper, we provide partial answers to the above open problems.