Phase transitions for induced-H-free graphon variational problems

Determine whether, for every graph H with coloring number at least three, the variational problem over induced-H-free graphons exhibits a phase transition as the edge density varies, and whether its optimizer is always nonunique in some parameter regime.

Background

The paper finds phase transitions and regimes with multiple optimizers for the fixed-edge-density variational problem over claw-free graphons, corresponding to induced-K1,3K_{1,3}-free graphs. It proposes investigating whether these phenomena persist for other induced-H-free hereditary classes.

The question specifically concerns graphs H whose coloring number is at least three, using coloring number as the appropriate induced-setting analogue of chromatic number.

References

For graphs $H$ with coloring number\footnote{The coloring number was defined by \citeauthor{promel1992excluding} and is the suitable replacement for chromatic number in the induced setting.} at least three, does the variational problem over induced-$H$-free graphons exhibit a phase transition in the edge density? Is there always a phase in which the optimizer is nonunique?

— The typical structure of dense claw-free graphs  (2501.17816 - Perkins et al., 29 Jan 2025) in Section 1, subsection “Future Work,” second displayed Question environment