Effect of removing the bounded-VC-dimension assumption

Determine whether removing the bounded-VC-dimension assumption changes the behavior of the asymmetric homomorphism threshold for pairs such as $(K_3,K_4)$, and in particular establish the unrestricted value of $\delta_{\mathrm{hom}}(K_3;K_4)$ when possible.

Background

The paper notes that the asymmetric threshold for (K3,K4)(K_3,K_4) has been resolved under the additional assumption that the host graph has bounded VC dimension. It is not known from the discussion whether the same behavior persists without that complexity restriction.

Resolving this issue would clarify whether bounded VC dimension is merely a technical hypothesis or whether it fundamentally affects asymmetric homomorphism thresholds.

References

We note that this question was resolved in Theorem 1.1 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 7, “Concluding remarks”