Test power-law asymptotics

Prove or disprove that for every graph G and graph family F there exist constants c=c(G,F) and d=d(G,F) such that φ(n,G|F)∼c n^d as n tends to infinity.

Background

Many results in the paper yield polynomial or Turán-type asymptotics, but the authors explicitly state that a universal power-law description is unknown.

References

It is not known whether every ø(n, G|F) is proportional to a real power of n at least in the asymptotic sense. Problem 10.8. Prove or disprove that for every graph G and family F of graphs there exist constants c = c(G, F) and d = d(G, F) such that ø(n, G|F) lim n+00 c . nd =1.

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.8, Section 10.1

It is not known whether every ø(n, G|F) is proportional to a real power of n at least in the asymptotic sense. Problem 10.8. Prove or disprove that for every graph G and family F of graphs there exist constants c = c(G, F) and d = d(G, F) such that ø(n, G|F) lim n+00 c . nd =1. (3)

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.8, Section 10.1