Anticoncentration for subgraph statistics

Abstract

Consider integers $k,\ell$ such that $0\le \ell \le \left(\genfrac{}{}{0ex}{}{k}{2}\right)$. 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 $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$, 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 ${\ell }^{\ast }=\min \{\ell ,(\genfrac{}{}{0ex}{}{k}{2})-\ell \}$. Our main result is that for any $k$ and $\ell$, the fraction of $k$-vertex subsets that span $\ell$ edges is at most ${\log }^{O\left(1\right)}\left({\ell }^{\ast }/k\right)\sqrt{k/{\ell }^{\ast }}$, 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.

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Cited by in corpus (12)

• Combinatorial anti-concentration inequalities, with applications
• A Completion of the Proof of the Edge-statistics Conjecture
• An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs
• The smallest singular value of random combinatorial matrices
• The feasible region of induced graphs
• The edge-statistics conjecture for $\ell \ll k^{6/5}$
• Majority dynamics on sparse random graphs
• Majority Dynamics: The Power of One
• Anticoncentration in Ramsey graphs and a proof of the Erdős-McKay conjecture
• Proof of a conjecture on induced subgraphs of Ramsey graphs
• An upper bound on the smallest singular value of dense random combinatorial matrices
• Inducibility in $H$-free graphs and inducibility of Turán graphs