Alon–Hefetz–Krivelevich–Tyomkyn edge-statistics conjecture

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

$$I(n,k,\ell):=\max\{\mathbb{P}[X_{G,k}=\ell]:v(G)=n\},$$

and

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

Edge-statistics conjecture. For all $k,\ell\in\mathbb{N}$ with $0<\ell<\binom{k}{2}$, we have

$$\operatorname{ind}(k,\ell)\leqslant 1/e+o_k(1).$$

The quantity $\operatorname{ind}(k,\ell)$ measures the largest asymptotic fraction of $k$-vertex subsets inducing exactly $\ell$ edges. The paper proves this bound in the range $1\leqslant\ell\leqslant o_k(k^{6/5})$, while the general assertion is presented as a conjecture originating with Alon, Hefetz, Krivelevich, and Tyomkyn.