Characterize positivity of the piercing function

Determine a necessary and sufficient condition for g(n,G|F)>0.

Background

The paper gives elementary necessary and sufficient conditions for positivity in Remark 2.4, but does not obtain a complete characterization. This problem asks for such a characterization.

References

Problem 10.3. Determine a condition which is necessary and sufficient for g(n,G|F)>0. (Cf. Remark 2.4.)

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.3, Section 10.1