Strictness of finite open chromatic number versus bounded cliques

Determine whether finite open chromatic number is strictly stronger than the bounded clique property for topological spaces.

Background

The paper defines finite open chromatic number by requiring every closed orthogonality relation to admit a finite cover by open orthogonality-free sets. It establishes the implication chain compactness implies finite open chromatic number implies bounded cliques.

Compactness is shown not to follow from finite open chromatic number, but the paper leaves unresolved whether the second implication is strict—namely, whether a space can have bounded cliques without having finite open chromatic number.