Banach Spaces and Ramsey Theory — Problem 22

Definitions and mathematical statement

Let $\omega_1$ denote the first uncountable ordinal, and let $[\omega_1]^{<\omega}$ denote the set of its finite subsets. A family $\mathcal F\subseteq[\omega_1]^{<\omega}$ is hereditary if $A\in\mathcal F$ and $B\subseteq A$ imply $B\in\mathcal F$. For the formulation below, identify each subset $A\subseteq\omega_1$ with its characteristic function in $\{0,1\}^{\omega_1}$, equipped with the product topology of the discrete topology on $\{0,1\}$. Compactness of $\mathcal F$ means compactness in this topology. This conventional topological interpretation is not explicitly specified in the supplied statement.

The question attributed to D. H. Fremlin is whether there exists a compact hereditary family $\mathcal F\subseteq[\omega_1]^{<\omega}$ such that \[ \{\alpha\}\in\mathcal F\qquad(\alpha<\omega_1) \] and \[ \forall A\in[\omega_1]^{<\omega}\ \exists B\subseteq A\quad \left(B\in\mathcal F\ \text{and}\ 2|B|\ge |A|\right). \] Here the source phrase "half of its size" is interpreted as "at least half", so the required size is at least $\lceil |A|/2\rceil$. Heredity then permits choosing $B$ of exactly that size. The source does not explicitly state the rounding convention for odd cardinalities; this interpretation requires review. For $A=\varnothing$, take $B=\varnothing$, which belongs to $\mathcal F$ by heredity and the singleton condition.

Short English statement

Does there exist a compact hereditary family of finite subsets of $\omega_1$, containing every singleton, such that every finite set contains a member of the family of at least half its cardinality? This draft uses the characteristic-function product topology and rounds half upward; both conventions require confirmation from the source context.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800757$, exact source problem code catalogue entry, title Banach Spaces and Ramsey Theory --- Problem 22. The registry labels the problem open; this supplied status has not been independently verified.
• Dodos and Lopez-Abad, Banach Spaces and Ramsey Theory: Some Open Problems (2010), Problem 22. Supplied source URL: https://arxiv.org/abs/1006.2668. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source consultation.
• The supplied statement attributes the question to D. H. Fremlin, with reference label [Fr2]. No bibliographic entry identifying [Fr2] is supplied, so the original reference cannot be identified precisely from this record.