Work

Maximum Shattering

Year: 2025

Type: article

Abstract: ABSTRACT A family of subsets of shatters a set if for every , there is an such that . We develop a framework to analyze , the maximum possible number of subsets of of size that can be shattered by a family of size . Among other results, we determine exactly for and show that if and grow, with both and tending to infinity, then for any satisfying , we have , where , roughly 0.289, is the probability that a large square matrix over is invertible. This latter result extends work of Das and Mészáros. As an application, we improve bounds for the existence of covering arrays for certain alphabet sizes.

Source: Journal of Combinatorial Designs

DOI: 10.1002/jcd.22005

Authors: Noga Alon, Varun Sivashankar, Daniel G. Zhu

Institution: Princeton University

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,350

Funder: National Science Foundation