$\mathsf{S}\mathsf{O}\mathsf{C}\mathsf{A}$ and $\mathsf{O}\mathsf{G}\mathsf{A}$ for $\mathsf{H}\mathsf{L}$ spaces with strong properties

We study open colorings in certain classes of hereditary Lindel"{o}f ($\mathsf{H}\mathsf{L}$) spaces and submetrizable spaces. In particular, we show that the definible version for the Open Graph Axiom ($\mathsf{O}\mathsf{G}\mathsf{A}$) holds for the class of $\mathsf{H}\mathsf{L}$ 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 $\mathsf{O}\mathsf{G}\mathsf{A}$”.