# Upper tail bounds for Stars

## TL;DR

This paper derives nearly optimal exponential bounds for the upper tail probabilities of the number of r-armed stars in a binomial random graph, extending previous results to a broader range of deviations.

## Contribution

It establishes the best possible exponential bounds for upper tail probabilities of star counts in G_{n,p}, solving a problem posed by Janson and Rucinski and confirming a conjecture by DeMarco and Kahn.

## Key findings

- Derived exponential bounds are tight up to constant factors.
- Extended upper tail analysis to deviations larger than constant .
- Confirmed conjecture and solved open problem for star counts.

## Abstract

For r \ge 2, let X be the number of r-armed stars K_{1,r} in the binomial random graph G_{n,p}. We study the upper tail \Pr(X \ge (1+\epsilon)\E X), and establish exponential bounds which are best possible up to constant factors in the exponent (for the special case of stars K_{1,r} this solves a problem of Janson and Rucinski, and confirms a conjecture by DeMarco and Kahn). In contrast to the widely accepted standard for the upper tail problem, we do not restrict our attention to constant \epsilon, but also allow for \epsilon \ge n^{-\alpha} deviations.