Determine extremal families for a feasible pair

Given a graph G and real numbers c>0 and 0≤d≤2, determine the inclusionwise minimal and maximal graph families F for which G and F determine the feasible pair (c,d).

Background

Monotonicity implies that families nested between two families producing the same asymptotics also produce that pair. The authors ask for the extremal families realizing a prescribed asymptotic pair.

References

Problem 10.18. Given a graph G and the real numbers c, d with c > 0 and 0sd≤ 2, find the inclusionwise minimal and maximal families F such that G and F determine (c,d).

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.18, Section 10.1