Extension of asymmetric homomorphism thresholds beyond odd-cycle families

Determine how to extend the asymmetric homomorphism-threshold results proved for families of odd cycles to more general forbidden-graph settings.

Background

The paper's threshold results rely heavily on the structure-preserving properties of generalized Mycielskian constructions and are developed specifically for odd-cycle families.

The authors explicitly identify the lack of a clear extension mechanism as an unresolved issue for broader asymmetric homomorphism-threshold problems.