{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:5HEQS23LLVGWLB6RFSPCYGUE5W","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d1ddd774b2715edd3a1f7c543067619bb998aca07fb9d2fbd206fe8317067693","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-11T05:25:37Z","title_canon_sha256":"eb1c172ef3dc68041a7782f660f9ff85264f608674d16b5efd7c2350d88f4337"},"schema_version":"1.0","source":{"id":"2508.07632","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.07632","created_at":"2026-07-05T11:51:49Z"},{"alias_kind":"arxiv_version","alias_value":"2508.07632v1","created_at":"2026-07-05T11:51:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.07632","created_at":"2026-07-05T11:51:49Z"},{"alias_kind":"pith_short_12","alias_value":"5HEQS23LLVGW","created_at":"2026-07-05T11:51:49Z"},{"alias_kind":"pith_short_16","alias_value":"5HEQS23LLVGWLB6R","created_at":"2026-07-05T11:51:49Z"},{"alias_kind":"pith_short_8","alias_value":"5HEQS23L","created_at":"2026-07-05T11:51:49Z"}],"graph_snapshots":[{"event_id":"sha256:2265db4f55902da0b9fe582cf4ef3dbf773737b0a64d79f09d06558f18f04833","target":"graph","created_at":"2026-07-05T11:51:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2508.07632/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an $n \\times n$ square lattice such that no $k+1$ of them are collinear. The problem was first posed more than $100$ years ago for the special case $k=2$ and has remained open ever since. The general problem was recently resolved in the case $k$ is not small compared to $n$, as Kov\\'acs, Nagy and Szab\\'o proved that the upper bound $kn$ can be attained, provided that $k>C\\sqrt{n\\log{n}}$ for an absolute constant $C$.\n  In this paper, we show that $\\left(1-\\tfrac{2}{k}\\right)kn \\leq f_k(n)\\leq kn$ and $\\le","authors_text":"Benedek Kov\\'acs, D\\'avid R. Szab\\'o, Zolt\\'an L\\'or\\'ant Nagy","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-11T05:25:37Z","title":"Randomised algebraic constructions for the no-$(k+1)$-in-line problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07632","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:39b0701c0e43e6262a1e2663378f9bcae955011117a8b6c84092941ce2701777","target":"record","created_at":"2026-07-05T11:51:49Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d1ddd774b2715edd3a1f7c543067619bb998aca07fb9d2fbd206fe8317067693","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-11T05:25:37Z","title_canon_sha256":"eb1c172ef3dc68041a7782f660f9ff85264f608674d16b5efd7c2350d88f4337"},"schema_version":"1.0","source":{"id":"2508.07632","kind":"arxiv","version":1}},"canonical_sha256":"e9c9096b6b5d4d6587d12c9e2c1a84eda3f4b70a8d67e051865c6dfaa6f1cfa0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e9c9096b6b5d4d6587d12c9e2c1a84eda3f4b70a8d67e051865c6dfaa6f1cfa0","first_computed_at":"2026-07-05T11:51:49.490990Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:51:49.490990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CJNS0BsCmn7gHNqlX/KBi7rHEBS5Tc4tEsV7Grz7BAk3n0CMegAKKaG7rQ2DWyBjl/Q0duqb7ULrYFLLQxCaDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:51:49.491531Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.07632","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:39b0701c0e43e6262a1e2663378f9bcae955011117a8b6c84092941ce2701777","sha256:2265db4f55902da0b9fe582cf4ef3dbf773737b0a64d79f09d06558f18f04833"],"state_sha256":"bcd91047b3aa8cf022a4cce3fa6cb8eaf09070bd60c5380de1a79a266ab046e0"}