{"paper":{"title":"On the discrepancy of random low degree set systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Nikhil Bansal, Raghu Meka","submitted_at":"2018-10-08T11:14:24Z","abstract_excerpt":"Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are $n$ elements and $m$ sets and each element lies in $t$ randomly chosen sets. In this setting, Ezra and Lovett showed an $O((t \\log t)^{1/2})$ discrepancy bound in the regime when $n \\leq m$ and an $O(1)$ bound when $n \\gg m^t$.\n  In this paper, we give a tight $O(\\sqrt{t})$ bound for the entire range of $n$ and $m$, under a mild assumption that $t = \\Omega (\\log \\log m)^2$. The result is based on two steps. First, applying the partial coloring method to the case when $n = m \\log^{O(1)} m$ and usin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03374","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"}