Alon–Hefetz–Krivelevich–Tyomkyn superlinear sparsity conjecture

For an $n$-vertex graph $G$, let $X_{G,k}$ be the number of edges induced by a uniformly random $k$-vertex subset of $G$. Define

$$I(n,k,\ell)=\max\{\Pr(X_{G,k}=\ell):|V(G)|=n\}$$

and

$$\operatorname{ind}(k,\ell)=\lim_{n\to\infty}I(n,k,\ell).$$

Alon–Hefetz–Krivelevich–Tyomkyn's superlinear sparsity conjecture. For all $k,\ell$ satisfying

$$\min\mathopen{}\mathclose\bgroup\originalleft\{\ell,\binom{k}{2}-\ell\aftergroup\egroup\originalright\}=\omega_k(k),$$

we have

$$\operatorname{ind}(k,\ell)=o_k(1).$$

This predicts asymptotic anticoncentration whenever both the number of induced edges and the number of nonedges grow superlinearly in $k$. Its status is not resolved in the supplied text.