$\mathsf{SOCA}$ and $\mathsf{OGA}$ for $\mathsf{HL}$ spaces with strong properties

TL;DR

This paper explores the validity of the Open Graph Axiom in hereditary Lindelöf and submetrizable spaces, extending known results and establishing consistency results related to a longstanding conjecture in topology.

Contribution

It proves that the definable version of OGA holds for HL strong Choquet submetrizable spaces and shows the consistency of OGA for certain regular spaces with countable spread.

Findings

OGA holds for HL strong Choquet submetrizable spaces

Consistency of OGA for regular spaces with countable spread

Progress towards Todorčević's conjecture

Abstract

We study open colorings in certain classes of hereditary Lindel\"{o}f ($\mathsf{HL}$) spaces and submetrizable spaces. In particular, we show that the definible version for the Open Graph Axiom ($\mathsf{OGA}$) holds for the class of $\mathsf{HL}$ strong Choquet submetrizable spaces extending a well-known result of Feng. Furthermore, we show the consistency of the Open Graph Axiom for regular spaces that have countable spread and it's square also has it, reaching closer to a well known conjecture of Todor\v{c}evi\'{c}: "It is consistent that all regular spaces with countable spread satisfy $\mathsf{OGA}$".