Anticoncentration for subgraph statistics

Consider integers k , ℓ such that 0 ⩽ ℓ ⩽ k 2 . Given a large graph G, what is the fraction of k-vertex subsets of G which span exactly ℓ edges? When G is empty or complete, and ℓ is zero or k 2 , this fraction can be exactly 1. On the other hand, if ℓ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich, and Tyomkyn who initiated the systematic study of this question and proposed several natural conjectures. Let ℓ ∗ = min { ℓ , k 2 − ℓ } . Our main result is that for any k and ℓ, the fraction of k-vertex subsets that span ℓ edges is at most log O ( 1 ) ( ℓ ∗ / k ) k / ℓ ∗ , which is best-possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich, and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towards some analogous questions for hypergraphs. Our proofs involve some Ramsey-type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a ‘slice’ of the Boolean hypercube.