Abstract Let r, k, be integers such that 0 kr. Given a large r-uniform hypergraph G, we consider the fraction of k-vertex subsets that span exactly edges. If is 0 or kr, 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 kr, then this fraction is at most (1/e) +, assuming k is sufficiently large in terms of r and 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–Trujić). 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 kr.
Jain et al. (Mon,) studied this question.
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