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Let $k$ and $\ell$ be positive integers. We prove that if $1 \leq \ell \leqo_k(k^{6/5})$, then in every large enough graph $G$, the fraction of $k$-vertexsubsets that induce exactly $\ell$ edges is at most $1/e + o_k(1)$. Togetherwith a recent result of Kwan, Sudakov, and Tran, this settles a conjecture ofAlon, Hefetz, Krivelevich, and Tyomkyn.

The edge-statistics conjecture for $\ell \ll k^{6/5}$