Extension beyond odd-cycle homomorphism thresholds

Develop asymmetric homomorphism-threshold results for graph families and graph pairs beyond the odd-cycle settings treated in the paper, including a method that does not depend on the specific structure of generalized Mycielskian constructions.

Background

The principal threshold results concern exclusions of consecutive odd cycles and rely on generalized Mycielskian constructions. The authors state that their techniques are highly specific to these families.

A concrete motivating example is the asymmetric threshold $\delta_{\mathrm{hom}}(K_3;K_4)$, as well as asymmetric thresholds for general pairs of cliques. The paper notes that bounded-VC-dimension cases are known, while the unrestricted setting remains unresolved.