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In particular, we show the existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1)) q^{\\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at most $O(q^{\\frac{n-e+1}{e}})$ if there exist integers $d,e"},"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":"2211.04329","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-11-08T15:47:12Z","cross_cats_sorted":[],"title_canon_sha256":"3ce245c57f4aa149b8cc8bf4c8174bca458ace88327b199554ed75f9ddaf444f","abstract_canon_sha256":"abeb24a621ed8e52378d5799933f2e50f61305bc2a8fdf00cf4c84ab931ec19c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:15:13.454617Z","signature_b64":"sQX62dx8uKIDBH3bXeGGj6JLQaTaPCk7GcCOkNMGWJpXg7lVOotmhd4bE0OKv0PcBioAKy6L9N4yLsMy6QRaDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ec22219a67b0219a3b2ccd4c80530cd154116d1c31dbebe379ead93759f5dec","last_reissued_at":"2026-07-05T05:15:13.454138Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:15:13.454138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large $(k; r, s; n, q)$-sets in Projective Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ferdinand Ihringer, Jacques Verstra\\\"ete","submitted_at":"2022-11-08T15:47:12Z","abstract_excerpt":"A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q \\rightarrow \\infty$. In particular, we show the existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1)) q^{\\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at most $O(q^{\\frac{n-e+1}{e}})$ if there exist integers $d,e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.04329","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.04329/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2211.04329","created_at":"2026-07-05T05:15:13.454196+00:00"},{"alias_kind":"arxiv_version","alias_value":"2211.04329v2","created_at":"2026-07-05T05:15:13.454196+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.04329","created_at":"2026-07-05T05:15:13.454196+00:00"},{"alias_kind":"pith_short_12","alias_value":"P3BCEGNGPMBB","created_at":"2026-07-05T05:15:13.454196+00:00"},{"alias_kind":"pith_short_16","alias_value":"P3BCEGNGPMBBTI5S","created_at":"2026-07-05T05:15:13.454196+00:00"},{"alias_kind":"pith_short_8","alias_value":"P3BCEGNG","created_at":"2026-07-05T05:15:13.454196+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.22289","citing_title":"$(r,s)$-sets from Desarguesian ovoids","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU","json":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU.json","graph_json":"https://pith.science/api/pith-number/P3BCEGNGPMBBTI5SZTKMQBJQZU/graph.json","events_json":"https://pith.science/api/pith-number/P3BCEGNGPMBBTI5SZTKMQBJQZU/events.json","paper":"https://pith.science/paper/P3BCEGNG"},"agent_actions":{"view_html":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU","download_json":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU.json","view_paper":"https://pith.science/paper/P3BCEGNG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2211.04329&json=true","fetch_graph":"https://pith.science/api/pith-number/P3BCEGNGPMBBTI5SZTKMQBJQZU/graph.json","fetch_events":"https://pith.science/api/pith-number/P3BCEGNGPMBBTI5SZTKMQBJQZU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU/action/storage_attestation","attest_author":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU/action/author_attestation","sign_citation":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU/action/citation_signature","submit_replication":"https://pith.science/pith/P3BCEGNGPMBBTI5SZTKMQBJQZU/action/replication_record"}},"created_at":"2026-07-05T05:15:13.454196+00:00","updated_at":"2026-07-05T05:15:13.454196+00:00"}