Determine piercing by matchings

Determine g(n,K4|2K2), and more generally determine g(n,K2p|pK2) and g(n,Kp|qK2) for p>2q.

Background

Exact piercing by 2K2 and broader matching families is only partially resolved. The paper asks for both the first unresolved case and generalizations.

References

Problem 10.22. Determine g(n, K4|2K2), and more generally g(n, K2ppK2) and beyond, g(n, KplqK2) where p > 2q.

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.22, Section 10.2