Countably compact spaces with finite but unbounded cliques

Construct a countably compact $T_1$ or Hausdorff space carrying a closed orthogonality relation whose cliques are all finite but have unbounded finite cardinalities.

Background

The paper establishes that, for T1T_1 spaces, the finite clique property is equivalent to countable compactness, while compactness implies the stronger bounded clique property. The unresolved issue is whether the second implication is strict: whether countable compactness can coexist with finite cliques of arbitrarily large size for a closed orthogonality relation.

A positive example would necessarily have a highly nonclassical structure. Later results show that it may be assumed separable, but it must be noncompact, nonsquentially compact, non-ω\omega-bounded, nonmetacompact, not uu-compact for any free ultrafilter, and must have a countable power that is not countably compact.

References

Whether the second implication is strict is the main open question of this note: Is there a countably compact $T_1$ (or Hausdorff) space with a closed orthogonality relation having finite but unbounded cliques?

— A Note on Compactness and Clique Size  (2608.13320 - Feldman et al., 13 Aug 2026) in Question 1, Section 1

However, we do not know whether, in the general, non-$T_1$ setting, $\omega$-lp compactness is necessary for finite cliques. The sufficiency of $\omega$-lp compactness gives us a non-$T_1$ version of Question 1: is there an $\omega$-lp compact space hosting a closed, irreflexive relation with cliques of arbitrary size?

— A Note on Compactness and Clique Size  (2608.13320 - Feldman et al., 13 Aug 2026) in Remark 2, Section 1

We expect the existence of such a $Y$ to be sensitive to set theory, consistently with the profile in Remark \ref{profile}; we leave it open, in ZFC and under CH or MA.

— A Note on Compactness and Clique Size  (2608.13320 - Feldman et al., 13 Aug 2026) in Section 4, construction following Proposition 4.2 and the concluding paragraph of the supplied text