{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2H4A4YZQQ2JF2OXGPLVBY7MX5I","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":"be7c492c75ad818acc60b039cf63af772c019507b449d7d814508f2cb899aa92","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:57:34Z","title_canon_sha256":"d19fe2f7c843c86872332e4a540dcc8ee8043c735e31e6c99770c164762698d2"},"schema_version":"1.0","source":{"id":"2607.15269","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15269","created_at":"2026-07-17T01:22:17Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15269v1","created_at":"2026-07-17T01:22:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15269","created_at":"2026-07-17T01:22:17Z"},{"alias_kind":"pith_short_12","alias_value":"2H4A4YZQQ2JF","created_at":"2026-07-17T01:22:17Z"},{"alias_kind":"pith_short_16","alias_value":"2H4A4YZQQ2JF2OXG","created_at":"2026-07-17T01:22:17Z"},{"alias_kind":"pith_short_8","alias_value":"2H4A4YZQ","created_at":"2026-07-17T01:22:17Z"}],"graph_snapshots":[{"event_id":"sha256:550422f6803050cddf64428f7ccff0e27de8d0cc6ee7e546c23e83372d5d929e","target":"graph","created_at":"2026-07-17T01:22:17Z","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/2607.15269/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove, for all fixed $0 < \\delta < 1$, and all sufficiently large $n$, that there exists $S \\subset [n]$ with $|S| \\ge \\delta n$ such that $A + B \\not \\subset S$ for all ${A, B \\subset \\mathbb{N}}$ satisfying $$\\min\\big\\{|A|, |B|\\big\\} \\ge \\big(3 + o(1)\\big) \\frac{\\log n }{ \\log (1 / \\delta)}.$$ A very recent result of Hern\\'andez and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $\\delta$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and ","authors_text":"Gabriel Dahia, Jo\\~ao Pedro Marciano, Victor Souza","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:57:34Z","title":"Dense sets without large sumsets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15269","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":"sha256:65c921ea52e9311d6a69f4e91d492ddd927dfef67ffd68929b6022fedca0a67d","target":"record","created_at":"2026-07-17T01:22:17Z","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":"be7c492c75ad818acc60b039cf63af772c019507b449d7d814508f2cb899aa92","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-16T17:57:34Z","title_canon_sha256":"d19fe2f7c843c86872332e4a540dcc8ee8043c735e31e6c99770c164762698d2"},"schema_version":"1.0","source":{"id":"2607.15269","kind":"arxiv","version":1}},"canonical_sha256":"d1f80e633086925d3ae67aea1c7d97ea2ac043e08304c1ac559f1b1b1b8feb68","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d1f80e633086925d3ae67aea1c7d97ea2ac043e08304c1ac559f1b1b1b8feb68","first_computed_at":"2026-07-17T01:22:17.738142Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-17T01:22:17.738142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"quBepNPj2VuAIFTgsJqiMMIz14M3jrIQDOCdFzvS6GDValINnenyauFTeqwnWzTR4mSll0e/eB5pw8nXGJWSCA==","signature_status":"signed_v1","signed_at":"2026-07-17T01:22:17.738994Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.15269","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65c921ea52e9311d6a69f4e91d492ddd927dfef67ffd68929b6022fedca0a67d","sha256:550422f6803050cddf64428f7ccff0e27de8d0cc6ee7e546c23e83372d5d929e"],"state_sha256":"2a9b91c76f160b5a92589b37aeb057de3b56766c0cbdcf76bb4282a16ac5175c"}