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For given integers $k$ and $\ell$ with $0<\ell< {k \choose 2}$, Alon, Hefetz,Krivelevich and Tyomkyn formulated the following conjecture: When sampling a$k$-vertex subset uniformly at random from a very large graph $G$, then theprobability to have exactly $\ell$ edges within the sampled $k$-vertex subsetis at most $e^{-1}+o_k(1)$. This conjecture was proved in the case$\Omega(k)\leq \ell\leq {k \choose 2}-\Omega(k)$ by Kwan, Sudakov and Tran. Inthis paper, we complete the proof of the conjecture by resolving the remainingcases. We furthermore give nearly tight upper bounds for the probabilitydescribed above in the case $\omega(1)\leq \ell\leq o(k)$. We also extend someof our results to hypergraphs with bounded edge size.

A Completion of the Proof of the Edge-statistics Conjecture