Cycle piercing in the intermediate range

Determine the asymptotics of g(n,K_p|C_q) for q<p≤2q−2, in particular resolving the cases (p,q)=(5,4) and (p,q)=(6,4).

Background

The paper obtains bounds for cycle piercing in the intermediate range using bipartite Turán numbers, but does not determine the exact asymptotic order in general. Two smallest unresolved cases are singled out.

References

Problem 10.24. Determine the asymptotics of g(n, Kpl(Ca)) for p ≤ 2q-2. In particular, settle the cases (p,q) = (5,4) and (p,q) = (6,4).

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.24, Section 10.2