Effect of bounded-VC-dimension assumptions on clique homomorphism thresholds

Determine whether removing the bounded-VC-dimension assumption from the known result for the asymmetric homomorphism threshold $\delta_{\hom}(K_3;K_4)$ changes its behavior.

Background

The paper identifies asymmetric homomorphism thresholds for pairs of cliques as a broader direction beyond its odd-cycle results, mentioning δhom⁡(K3;K4)\delta_{\hom}(K_3;K_4) as a basic example.

A cited result resolves the question under the additional assumption that the host graph has bounded VC dimension. The unresolved issue is whether the threshold behaves differently without that structural assumption.

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 5, “Concluding remarks”