Generalisations of \(\varepsilon\)-density. (English) Zbl 1057.03031

Džamonja, Mirna; Plebanek, Grzegorz

Acta Univ. Carol., Math. Phys. 44, No. 2, 57-64 (2003).

Summary: We give several partial solutions to Fremlin’s question DU about the existence of large homogeneous sets for \(\varepsilon\)-dense open families of finite sets of ordinals by introducing and considering some generalisations of the notion of \(\varepsilon\)-density. In particular we prove that every \(\frac 12\)-functionally dense open family has an infinite homogeneous set and that under \(\text{MA}+\neg\text{CH}\) every \(\frac 12\)-dense open family satisfying an additional covering property is \(\frac 12\)-functionally dense and has a homogeneous set of size \(\aleph_1\). Moreover, we prove that assuming \(\text{MA}+\neg\text{CH}\) satisfying this covering property on a set of size \(\aleph_1\), and that under the same assumptions functional density on a set of size \(\aleph_1\) is another necessary and sufficient condition for such a homogeneous set to exist.
We also study the continuous version of \(\varepsilon\)-density and give some negative homogeneity results.

MSC:

03E02	Partition relations
03E50	Continuum hypothesis and Martin’s axiom
28C99	Set functions and measures on spaces with additional structure

Keywords:

large homogeneous sets; \(\varepsilon\)-density; dense open family; covering property

Citations:

Zbl 1021.03046