Abstract:

Almost counterexamples

Say that a family F of subsets of a set A is 1/2-dense if every
finite F\subseteq A has a subset F_0 in F satisfying |F_0| >= 1/2
|F|. The notorious problem DU on Fremlin's list is if every
1/2-dense family on \omega_1 which is closed under subsets admits
an infinite set whose all finite subsets are in F. Under CH the
answer is known to be negative.

In the hope of getting closer to the solution of DU we try to
understand the relevance of the two requirements on the family. We
show that there is a 1/2-dense family F of subsets of the
continuum c with the property that every infinite subset of  c has
arbitrarily large finite subsets in F but there is no infinite set
whose all finite subsets are in F. Michalewski found an easy
example example which closed under subsets but where 1/2-density
is weakened. By a result of Fremlin, modulo a measurable cardinal
it is consistent that for every 1/2-dense closed under subsets
family on c there is an infinite set whose all finite subsets are
in the family, therefore the above examples are closest that one
can get to a counterexample in ZFC.