A Completion of the Proof of the Edge-statistics Conjecture

For given integers $k$ and $\mathrm{\ell }$ with $0<\mathrm{\ell }<\left(\genfrac{}{}{0px}{}{k}{2}\right)$, 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 the probability to have exactly $\mathrm{\ell }$ edges within the sampled $k$-vertex subset is at most $e^{-1}+o_k(1)$. This conjecture was proved in the case $\mathrm{\Omega }(k)\le \mathrm{\ell }\le \left(\genfrac{}{}{0px}{}{k}{2}\right)-\mathrm{\Omega }(k)$ by Kwan, Sudakov and Tran. In this paper, we complete the proof of the conjecture by resolving the remaining cases. We furthermore give nearly tight upper bounds for the probability described above in the case $\omega (1)\le \mathrm{\ell }\le o(k)$. We also extend some of our results to hypergraphs with bounded edge size.