On the Beck-Fiala conjecture for random set systems

Esther Ezra, Shachar Lovett

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Σ), where each element x ∈ X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X,Σ) is with high probability (Formula presented.); and when |X| ≫ |Σ|t the hereditary discrepancy of (X,Σ) is with high probability O(1). The first bound combines the Lovász Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.

Original languageEnglish
Pages (from-to)665-675
Number of pages11
JournalRandom Structures and Algorithms
Volume54
Issue number4
DOIs
StatePublished - Jul 2019

Bibliographical note

Publisher Copyright:
© 2018 Wiley Periodicals, Inc.

Keywords

  • beck-fiala conjecture
  • discrepancy theory
  • random set systems

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