Characterize eventual 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 pair (k,0) is feasible.

Background

The paper establishes eventual constancy whenever the relevant function is bounded, but does not classify the possible eventual constants or determine whether every positive integer occurs.

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