{"paper":{"title":"On the Beck-Fiala Conjecture for Random Set Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Esther Ezra, Shachar Lovett","submitted_at":"2015-11-02T16:59:15Z","abstract_excerpt":"Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,\\Sigma)$, where each element $x \\in X$ lies in $t$ randomly selected sets of $\\Sigma$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We show that when $|\\Sigma| \\ge |X|$ the hereditary discrepancy of $(X,\\Sigma)$ is with high probability $O(\\sqrt{t \\log t})$; and when $|X| \\gg |\\Sigma|^t$ the hereditary discrepancy of $(X,\\Sigma)$ is with high probability $O(1)$. The first bound combines the Lov{\\'a}sz Local Lemma with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00583","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}