Subquadratic growth for generalized decompositions

Characterize the families of graphs that are minimal under inclusion and for which the function f(n,G|F) is subquadratic for every graph G.

Background

The function f(n,G|F) measures the number of colors needed to force a copy of G whose monochromatic color classes belong to F. The paper establishes examples with constant, subquadratic, and quadratic growth, but does not identify the precise family-theoretic threshold for universal subquadratic behavior.

References

Problem 8.1. Characterize the families F minimal under inclusion, for which f(n, G|F) is subquadratic for every G.

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 8.1, Section 8