{"paper":{"title":"A note on piercing discrete rectangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wei Rao","submitted_at":"2026-04-05T08:45:03Z","abstract_excerpt":"In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in $\\mathbb R^d$. Very recently, this result was extended to the $(p,q)$ setting with $p \\geq q \\geq d+1$ by Edwards and Sober\\'on, and subsequently to the case $p \\geq q \\geq 2$ by Gangopadhyay, Polyanskii, and the author of this paper.\n  In this paper, we obtain improved bounds for the $(p,q)$ problem in the case $q=2$ and $d=2$. More precisely, our main result asserts that for any integer $p \\geq 2$, any set $P \\subseteq \\mathbb R^2$, and any finite family $\\mathcal B$ of axis-parallel rectangles in $\\mathbb R^2$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.04024","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.04024/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}