Path piercing in the intermediate range

Determine the asymptotics of g(n,K_p|P_q) for q<p≤2q−2.

Background

Theorem 9.26 settles the quadratic regime p≥2q−1, while the intermediate range is only partially bounded. The authors explicitly leave its asymptotic behavior unresolved.

References

Problem 10.23. Determine the asymptotics of g(n, Kpl(Pa)) for p≤ 2q-2.

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.23, Section 10.2