Compact 1/2-filling family on omega-one

Construct a compact hereditary family \(\mathcal{F}\subseteq [\omega_1]^{<\omega}\) such that, for every finite set \(A\subseteq\omega_1\), there is an \(F\in\mathcal{F}\) with \(|F|>|A|/2\).

Background

The paper discusses attempts to generalize the Schreier family from countable combinatorial Banach spaces to the uncountable cardinal ω1\omega_1. A compact hereditary family on ω1\omega_1 with the stated 1/2-filling property would provide such an analogue and could be used to define a new class of nonseparable combinatorial Banach spaces. The problem is attributed to Fremlin.

References

It is worth mentioning that there is a famous open problem if one can define a family of properties similar to Schreier family on \omega_1: Fremlin in asked if there is a family \mathcal{F}\subseteq [\omega_1]{<\omega}} which is compact and $1/2$-filling, i.e. such that for each $A\in [\omega_1]{<\omega}$ there is $F\in \mathcal{F}$ such that $|F|>|A|/2$.

— Non-separable combinatorial Banach spaces  (2610.01919 - Borodulin-Nadzieja et al., 1 Oct 2026) in Section 5, Examples, Example [compact families]