Alon–Bollobás–Krivelevich–Sudakov conjecture on Max-Cut in H-free graphs

Let $H$ be a graph. A graph is $H$-free if it contains no subgraph isomorphic to $H$, and for a graph $G$ let $f(G)$ denote the maximum number of edges in a cut of $G$. Let $m\geq 1$ be the number of edges of $G$.

Alon–Bollobás–Krivelevich–Sudakov conjecture. There exist constants $\varepsilon=\varepsilon(H)>0$ and $c=c(H)>0$ such that, for all $H$-free graphs $G$,

$$f(G)\geq \frac{m}{2}+cm^{3/4+\varepsilon}.$$

This conjecture is presented as a closely related consequence or target of the preceding degeneracy-based conjecture and concerns Max-Cut lower bounds depending only on the edge count. Its general validity is not established in the source.