Anticoncentration for subgraph statistics
Abstract
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 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 intiated the systematic study of this question and proposed several natural conjectures. Let $\ell^{*}=\min\{\ell,\binom{k}{2}\ell\}$. Our main result is that for any $k$ and $\ell$, the fraction of $k$vertex subsets that span $\ell$ edges is at most $\log^{O\left(1\right)}\left(\ell^{*}/k\right)\sqrt{k/\ell^{*}}$, which is bestpossible 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 Ramseytype 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.05202
 Bibcode:
 2018arXiv180705202K
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 doi:10.1112/jlms.12192