Asymmetric clique homomorphism threshold without bounded VC dimension

Determine whether removing the bounded-VC-dimension assumption changes the value or behavior of the asymmetric homomorphism threshold for $\delta_{\hom}(K_3;K_4)$.

Background

The paper identifies the asymmetric homomorphism threshold for a pair of cliques, especially δhom⁡(K3;K4)\delta_{\hom}(K_3;K_4), as an unresolved direction. A cited result resolves the question under the additional assumption that the host graph has bounded VC dimension.

It remains unknown whether the unrestricted problem has the same behavior as the bounded-complexity case, so the open issue is specifically whether bounded VC dimension is essential.

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”