Characterize constant small piercing values

Characterize the pairs (G,F) for which g(n,G|F)=1 or, more generally, g(n,G|F)=k for a small fixed integer k.

Background

Several exact constant values of the piercing function are established in the paper, but there is no general classification of pairs producing these small values.

References

Problem 10.4. Characterize the pairs (G, F) for which g(n, G|F) = 1, or a little further g(n, G|F) = k for k small.

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.4, Section 10.1