Sharp Concentration of Hitting Size for Random Set Systems

TL;DR

This paper investigates the minimal size of hitting sets in a random set system, demonstrating sharp concentration results using the second moment method across different probabilities p.

Contribution

It applies the second moment method to establish sharp concentration of the minimal hitting set size for various p values in a random set system.

Findings

Minimal hitting set size concentrates sharply for different p values.

The second moment method effectively analyzes the distribution of hitting set sizes.

Results provide insights into the structure of random set systems.

Abstract

Consider the random set system of {1,2,...,n}, where each subset in the power set is chosen independently with probability p. A set H is said to be a hitting set if it intersects each chosen set. The second moment method is used to exhibit the sharp concentration of the minimal size of H for a variety of values of p.