The edge-statistics conjecture for hypergraphs

Let r, k, be integers such that 0 ≤ ≤ (k/r). Given a large r-uniform hypergraph G, we consider the fraction of k-vertex subsets that span exactly edges. If is 0 or (k/r), this fraction can be exactly 1 (by taking G to be empty or complete), but for all other values of , one might suspect that this fraction is always significantly smaller than 1. In this paper we prove an essentially optimal result along these lines: if is not 0 or (k/r), then this fraction is at most (1/e) + ε, assuming k is sufficiently large in terms of r and ε > 0, and G is sufficiently large in terms of k. Previously, this was only known for a very limited range of values of r, k, (due to Kwan–Sudakov–Tran, Fox–Sauermann, and Martinsson–Mousset–Noever–Trujic). Our result answers a question of Alon–Hefetz–Krivelevich–Tyomkyn, who suggested this as a hypergraph generalization of their edge-statistics conjecture. We also prove a much stronger bound when is far from 0 and (k/r).