Alon–Bollobás–Krivelevich–Sudakov surplus conjecture for H-free graphs

Let $G$ be a graph with $m$ edges, and let $\mathrm{sp}(G)=\mathrm{mc}(G)-m/2$ denote its surplus, where $\mathrm{mc}(G)$ is the number of edges in a largest bipartite subgraph of $G$. For a fixed graph $H$, let $\mathrm{sp}(m,H)$ be the smallest value of $\mathrm{sp}(G)$ over all $H$-free graphs $G$ with $m$ edges. Alon–Bollobás–Krivelevich–Sudakov conjecture. For any fixed graph $H$, there is a constant $\epsilon(H)>0$ such that

$$\mathrm{sp}(m,H)=\Omega(m^{3/4+\epsilon(H)}).$$

The conjecture predicts a uniform improvement over the general $\Omega(\sqrt m)$ surplus bound for every fixed forbidden graph $H$. It remains open; the best known lower bounds establish weaker exponents for particular families of $H$.