{"paper":{"title":"Upper bounds on the smallest size of a saturating set in projective planes and spaces of even dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Alexander Davydov, Daniele Bartoli, Fernanda Pambianco, Massimo Giulietti, Stefano Marcugini","submitted_at":"2017-02-25T19:26:41Z","abstract_excerpt":"In a projective plane $\\Pi_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\\mathcal{S}$ is saturating (or dense) if any point of $\\Pi_{q}\\setminus \\mathcal{S}$ is collinear with two points in $\\mathcal{S}$. Modifying an approach of [31], we proved the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\\Pi_{q}$: \\begin{equation*} s(2,q)\\leq \\sqrt{(q+1)\\left(3\\ln q+\\ln\\ln q +\\ln\\frac{3}{4}\\right)}+\\sqrt{\\frac{q}{3\\ln q}}+3. \\end{equation*} The bound holds for all q, not necessarily large.\n  By using inductive constructions, upper bounds on the smallest "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.07939","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"}