{"paper":{"title":"On upper bounds on the smallest size of a saturating set in a projective plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander A. Davydov, Daniele Bartoli, Fernanda Pambianco, Massimo Giulietti, Stefano Marcugini","submitted_at":"2015-05-06T16:42:08Z","abstract_excerpt":"In a projective plane $\\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\\Pi _{q}\\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following upper bound on the smallest size $ s(2,q)$ of a saturating set in $\\Pi _{q}$ is proved: \\begin{equation*} s(2,q)\\leq 2\\sqrt{(q+1)\\ln (q+1)}+2\\thicksim 2\\sqrt{q\\ln q}. \\end{equation*} We also show that for any constant $c\\ge 1$ a random point set of size $k$ in $\\Pi _{q}$ with $ 2c\\sqrt{(q+1)\\ln(q+1)}+2\\le k<\\frac{q^{2}-1}{q+2}\\thicksim q$ is a saturating set"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01426","kind":"arxiv","version":3},"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"}