The Edge-statistics Conjecture for edge-inducibility

Let $k$ and $ll$ be integers with $0<ll<{k\choose 2}$. For $n\geq k$, let $I(n,k,ll)$ be the maximum, over all $n$-vertex graphs $G$, of the probability that a uniformly random $k$-vertex subset of $G$ spans exactly $ll$ edges, and define the edge-inducibility by

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

The Edge-statistics Conjecture. For all integers $k$ and $\ell$ with $0<\ell<{k\choose 2}$,

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

The conjecture gives a uniform asymptotic upper bound on the probability of observing any nontrivial prescribed number of edges in a random $k$-vertex subset of a very large graph. The paper states that earlier work proved substantial ranges and that the present result resolves the remaining cases, so the conjecture is solved.