{"paper":{"title":"A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Esther Ezra","submitted_at":"2013-07-30T20:28:22Z","abstract_excerpt":"Let $(X,\\S)$ be a set system on an $n$-point set $X$. The \\emph{discrepancy} of $\\S$ is defined as the minimum of the largest deviation from an even split, over all subsets of $S \\in \\S$ and two-colorings $\\chi$ on $X$. We consider the scenario where, for any subset $X' \\subseteq X$ of size $m \\le n$ and for any parameter $1 \\le k \\le m$, the number of restrictions of the sets of $\\S$ to $X'$ of size at most $k$ is only $O(m^{d_1} k^{d-d_1})$, for fixed integers $d > 0$ and $1 \\le d_1 \\le d$ (this generalizes the standard notion of \\emph{bounded primal shatter dimension} when $d_1 = d$). In th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.8139","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"}