Threshold Progressions in a Variety of Covering and Packing Contexts

TL;DR

This paper investigates threshold phenomena in covering and packing problems across combinatorics, revealing how coverage levels influence probabilistic thresholds in various mathematical contexts.

Contribution

It introduces a unified probabilistic framework to analyze threshold progressions in covering and packing problems across different combinatorial settings.

Findings

Threshold hierarchies depend on the level of coverage and dependence.

Results vary across extremal set theory, combinatorics, and additive number theory.

Standard probabilistic methods effectively analyze these threshold phenomena.

Abstract

Using standard methods (due to Janson, Stein-Chen, and Talagrand) from probabilistic combinatorics, we explore the following general theme: As one progresses from each member of a family of objects ${\cal A}$ being "covered" by at most one object in a random collection ${\cal C}$, to being covered at most $\lambda$ times, to being covered at least once, to being covered at least $\lambda$ times, a hierarchy of thresholds emerge. We will then see how such results vary according to the context, and level of dependence introduced. Examples will be from extremal set theory, combinatorics, and additive number theory.