Given integers $k,l$ and a graph $G$, how large can be the fraction of $k$-vertex subsets of $G$ which span exactly $l$ edges? The systematic study of this very natural question was recently initiated by Alon, Hefetz, Krivelevich and Tyomkyn who also proposed several interesting conjectures on this topic.
In this talk we discuss a theorem which proves one of their conjectures and implies an asymptotic version of another. We also make some first steps towards analogous question for hypergraphs. Our proofs involve some Ramsey-type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities and hypercontractivity.
Joint work with M. Kwan and T. Tran.