Eventually constant values

Given a positive integer c, characterize the pairs (G,F) for which φ(n,G|F)=c for all sufficiently large n, and determine whether every positive integer k occurs as such an eventual value.

Background

Theorem 10.13 establishes eventual constancy whenever the function is bounded. The authors ask which constants can occur and which graph-family pairs realize them.

References

Problem 10.15. Given an integer c > 0, characterize the pairs (G, F) for which $(n, G|F) = c for all n sufficiently large. In particular, is (k, 0) a feasible pair for every positive integer k ?

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.15, Section 10.1