Consider integers $k,\ell$ such that $0\le \ell \le \binom{k}2$. Given a large graph $G$, what is the fraction of $k$-vertex subsets of $G$ which span exactly $\ell$ edges? When $G$ is empty or complete, and $\ell$ is zero or $\binom k 2$, this fraction can be exactly 1. On the other hand if $\ell$ is not one these extreme values, then by Ramsey's theorem, this fraction is strictly smaller than 1.
The systematic study of the above question was recently initiated by Alon, Hefetz, Krivelevich and Tyomkyn, who proposed several natural conjectures. 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.