×

Some remarks on a question of D. H. Fremlin regarding \(\varepsilon\)-density. (English) Zbl 1021.03046

Summary: We show the relative consistency of \(\aleph_1\) satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of \(\omega_1\) satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of \(\omega_1\), and specifically if this and a stronger property hold under \(\operatorname {MA} + \neg \operatorname {CH}\). Towards this we show that if \(\operatorname {MA} + \neg \operatorname {CH}\) holds, then for every family \(\mathcal A\) of \(\aleph_1\) many infinite subsets of \(\omega_1\), one can find a family \(\mathcal S\) of finite subsets of \(\omega_1\) which is dense in Fremlins sense and does not contain all finite subsets of any set in \(\mathcal A\).
We then pose some open problems related to the question.

MSC:

03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
03E02 Partition relations
03E55 Large cardinals
03E45 Inner models, including constructibility, ordinal definability, and core models
Full Text: DOI