A counterexample related to a theorem of Komjath and Weiss

TL;DR

This paper presents a ZFC construction of a counterexample space that challenges a theorem by Komjath and Weiss, which previously relied on additional set-theoretic assumptions.

Contribution

The authors provide a ZFC example of a space with specific properties that contradicts a known theorem, removing the need for extra assumptions like iamondsuit.

Findings

Counterexample space constructed in ZFC

Space has size continuum and character 

Fails to satisfy the original theorem's conclusion

Abstract

In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak b$, if $X\rightarrow(top~\omega+1)^1_\omega$, then $X\rightarrow(top~\alpha)^1_\omega$ for all $\alpha<\omega_1$. In addition, assuming $\diamondsuit$, they constructed a space $X$ of size continuum, of character $\mathfrak b$, satisfying $X\rightarrow(top~\omega+1)^1_\omega$, but not $X\rightarrow(top~\omega^2+1)^1_\omega$. Here, a counterexample space with the same characteristics is obtained outright in ZFC.