On the Beck‐Fiala conjecture for random set systems. Issue 4 (7th November 2018) - Ezra, Esther; Lovett, Shachar
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 O ( t log t ) ; 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.