Characterize equality in the complementary relation

Determine a necessary and sufficient condition for g(n,G|F)=f(n,G|F)-1.

Background

Proposition 2.2 proves the equality under specific hypotheses involving family complementation and containment of G. The authors ask for a complete characterization of when equality holds.

References

Problem 10.5. Determine a condition which is necessary and sufficient for g(n, G|F) =f(n,G|F)-1. (Cf. Proposition 2.2.)

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.5, Section 10.1