Some remarks on a question of D. H. Fremlin regarding epsilon-density

We show the relative consistency of N-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 cardina l of uncountable cofinality can satisfy a similar, stronger property. The q uestions considered by D. H. Fremlin are if families of finite subsets of o mega (1) satisfying a certain density condition necessarily contain all fin ite subsets of an infinite subset of omega (1), and specifically if this an d a stronger property hold under MA + -CH. Towards this we show that if MA + -CH holds. then for every family. A of N-1 many infinite subsets of omega (1) one can find a family L of finite subsets of omega (1) which is dense in Fremlin's sense, and does not contain all finite subsets of any set in A . We then pose some open problems related to the question.