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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1702.07939","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-02-25T19:26:41Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"32f3b4aaf5e79990e74434501830b0005d06f4693b33372211c70d7d52b0ab37","abstract_canon_sha256":"22a87ca6cf2a8c30c616e6ebaeb18307a76db24651075bfd525f550e59a747dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:56.639064Z","signature_b64":"3Fy7Q/mI4I1SfizpsvjCwL3SJOoG+lryMiHt5TWJFmG87bcUAXBIL+4omhZQAypQmDC9LRTX+fmCT2wADDeIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5b1ad211370adc1ac93176e5db2e82b1f89bba1378c1d2c5130cf9e26ca4fe2","last_reissued_at":"2026-05-18T00:49:56.638423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:56.638423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1702.07939","created_at":"2026-05-18T00:49:56.638525+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.07939v1","created_at":"2026-05-18T00:49:56.638525+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.07939","created_at":"2026-05-18T00:49:56.638525+00:00"},{"alias_kind":"pith_short_12","alias_value":"YWY22IITOCW4","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_16","alias_value":"YWY22IITOCW4DLET","created_at":"2026-05-18T12:31:56.362134+00:00"},{"alias_kind":"pith_short_8","alias_value":"YWY22IIT","created_at":"2026-05-18T12:31:56.362134+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM","json":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM.json","graph_json":"https://pith.science/api/pith-number/YWY22IITOCW4DLETC5XF3MXIFM/graph.json","events_json":"https://pith.science/api/pith-number/YWY22IITOCW4DLETC5XF3MXIFM/events.json","paper":"https://pith.science/paper/YWY22IIT"},"agent_actions":{"view_html":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM","download_json":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM.json","view_paper":"https://pith.science/paper/YWY22IIT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.07939&json=true","fetch_graph":"https://pith.science/api/pith-number/YWY22IITOCW4DLETC5XF3MXIFM/graph.json","fetch_events":"https://pith.science/api/pith-number/YWY22IITOCW4DLETC5XF3MXIFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM/action/storage_attestation","attest_author":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM/action/author_attestation","sign_citation":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM/action/citation_signature","submit_replication":"https://pith.science/pith/YWY22IITOCW4DLETC5XF3MXIFM/action/replication_record"}},"created_at":"2026-05-18T00:49:56.638525+00:00","updated_at":"2026-05-18T00:49:56.638525+00:00"}