Deviation probabilities for arithmetic progressions and irregular discrete structures

TL;DR

This paper establishes bounds on the probability that the number of edges in a hypergraph induced by a random subset deviates from its mean, with applications to arithmetic progressions and irregular discrete structures.

Contribution

It introduces new probabilistic bounds for hypergraph edge counts in irregular settings, extending results to random vertex inclusion models and arithmetic progressions.

Findings

Bounds on deviation probabilities for hypergraph edge counts

Extension of arithmetic progression results to random subsets

Connections to central limit theorems in discrete structures

Abstract

Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$-element subset $B$ of its vertex set. Focussing on the case that the degrees of vertices in $\mathcal{H}$ vary significantly we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the $\{1,\dots, N\}$. Furthermore, our main theorem allows us to deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$. In this setting our result on arithmetic progressions extends a result of Bhattacharya, Ganguly, Shao and Zhao \cite{BGSZ}. We also mention connections to related central limit theorems.