Threshold Progressions in a Variety of Covering and Packing Contexts
Anant Godbole, Thomas Grubb, Kyutae Han, Bill Kay

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 being "covered" by at most one object in a random collection , to being covered at most times, to being covered at least once, to being covered at least 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.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Markov Chains and Monte Carlo Methods · Point processes and geometric inequalities
