Unrestricted asymmetric clique homomorphism threshold

Determine whether removing the bounded-VC-dimension assumption changes the asymmetric homomorphism threshold for the pair (K_3,K_4), and more generally determine asymmetric homomorphism thresholds for pairs of cliques without bounded-VC-dimension restrictions.

Background

The paper notes that the asymmetric threshold δ_hom(K_3;K_4) and the analogous problem for pairs of cliques are unresolved in the unrestricted setting discussed there.

A cited result resolves the question under the additional assumption that the host graph has bounded VC dimension; the open issue is whether that assumption is essential or changes the behavior.

References

We note that this question was resolved in \cite[Theorem 1.1]{2502.09576} under the added assumption that the host graph has bounded VC dimension, but it is not clear if removing this assumption would change the behaviour.

— Asymmetric results about graph homomorphisms  (2502.20278 - Gishboliner et al., 27 Feb 2025) in Section 6, Concluding remarks