{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HZLKWPX4VOJNEV43YV6MWHIWMS","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":"573556c8a075501b7b3bb767fdeb767399ea78c12a15fd3783cda0398a21473b","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-23T14:12:59Z","title_canon_sha256":"7ab38b07cc96f627d05cd4ece59a064d45ed1b7a8b330679ddbf13de8295f952"},"schema_version":"1.0","source":{"id":"1805.09188","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.09188","created_at":"2026-05-18T00:15:08Z"},{"alias_kind":"arxiv_version","alias_value":"1805.09188v1","created_at":"2026-05-18T00:15:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.09188","created_at":"2026-05-18T00:15:08Z"},{"alias_kind":"pith_short_12","alias_value":"HZLKWPX4VOJN","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HZLKWPX4VOJNEV43","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HZLKWPX4","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:9e05f95e64e1b0b2b9121bdf0a5f32905a220f3340be5eda160bca4d7d48998c","target":"graph","created_at":"2026-05-18T00:15:08Z","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"},"paper":{"abstract_excerpt":"We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $\\mathbb R^2$, with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is $O(n^{11/6})$, improving a result of Chang and Solymosi.\n  We also consider constructions for this problem. Alon, Ruzsa and Solymosi constructed an arrangement of four non-collinear $n$-pencils which determine $\\Omega(n^{3/2})$ four-rich points. We give a construction to show that this is not tight, improving ","authors_text":"Audie Warren, Oliver Roche-Newton","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-23T14:12:59Z","title":"Improved Bounds for Pencils of Lines"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.09188","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:a3e22b3c9d90c5526aadd8e96a39a39d04c6d41c6d37a782f71c0e2e9115341b","target":"record","created_at":"2026-05-18T00:15:08Z","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":"573556c8a075501b7b3bb767fdeb767399ea78c12a15fd3783cda0398a21473b","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-23T14:12:59Z","title_canon_sha256":"7ab38b07cc96f627d05cd4ece59a064d45ed1b7a8b330679ddbf13de8295f952"},"schema_version":"1.0","source":{"id":"1805.09188","kind":"arxiv","version":1}},"canonical_sha256":"3e56ab3efcab92d2579bc57ccb1d1664b46f9e34256114af0a2681b387c80ed9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e56ab3efcab92d2579bc57ccb1d1664b46f9e34256114af0a2681b387c80ed9","first_computed_at":"2026-05-18T00:15:08.262930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:08.262930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rHbBwd0xD/mbMQOWSdl+ic2IWuz+Ho6Bkx6xCQ02gTXHmhafd4wFfY/hLfa3KZZUq5ralamhhNbCDyL+y+Z4AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:08.263490Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.09188","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a3e22b3c9d90c5526aadd8e96a39a39d04c6d41c6d37a782f71c0e2e9115341b","sha256:9e05f95e64e1b0b2b9121bdf0a5f32905a220f3340be5eda160bca4d7d48998c"],"state_sha256":"e09e601d0962ef7ccf162e9d3afeff511645d61cb04e5fcf0d0b2878891d3061"}