{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:37SRE3NO73JQ6K3VBPTOY37GYW","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":"7be22713da37752f269cfd99623c69a7311e84b90d883a1c8746c3a3fc6ea7a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-06-17T17:40:31Z","title_canon_sha256":"138d9a729e9242dfd2cc2aa6ddbe3c60ea8bfb46fa784c8eb5261ee0c5c98197"},"schema_version":"1.0","source":{"id":"2106.09688","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.09688","created_at":"2026-06-23T03:13:42Z"},{"alias_kind":"arxiv_version","alias_value":"2106.09688v2","created_at":"2026-06-23T03:13:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.09688","created_at":"2026-06-23T03:13:42Z"},{"alias_kind":"pith_short_12","alias_value":"37SRE3NO73JQ","created_at":"2026-06-23T03:13:42Z"},{"alias_kind":"pith_short_16","alias_value":"37SRE3NO73JQ6K3V","created_at":"2026-06-23T03:13:42Z"},{"alias_kind":"pith_short_8","alias_value":"37SRE3NO","created_at":"2026-06-23T03:13:42Z"}],"graph_snapshots":[{"event_id":"sha256:f9192be7e62495ae3c48d555db5112fdfaa45a62bdd2a50c6f31ab68f67f4156","target":"graph","created_at":"2026-06-23T03:13:42Z","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/2106.09688/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\\in \\mathbb{N}$, the $r$-independence number of $G$, denoted $\\alpha_r(G)$, is the largest size of a $K_r$-free set of vertices in $G$. In this paper, we discuss Ramsey--Tur\\'an-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal $F$-tilings. For cliques, we show that for any $k\\geq 3$ and $\\eta>0$, any graph $G$ on $n$ vertices ","authors_text":"Donglei Yang, Guanghui Wang, Jie Han, Patrick Morris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-06-17T17:40:31Z","title":"A Ramsey-Tur\\'an theory for tilings in graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.09688","kind":"arxiv","version":2},"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":"sha256:6ff63c2da64afe540d350f16230bec09879b252dfe724854063bd6958823cad7","target":"record","created_at":"2026-06-23T03:13:42Z","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":"7be22713da37752f269cfd99623c69a7311e84b90d883a1c8746c3a3fc6ea7a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-06-17T17:40:31Z","title_canon_sha256":"138d9a729e9242dfd2cc2aa6ddbe3c60ea8bfb46fa784c8eb5261ee0c5c98197"},"schema_version":"1.0","source":{"id":"2106.09688","kind":"arxiv","version":2}},"canonical_sha256":"dfe5126daefed30f2b750be6ec6fe6c5b836011ce2ae230840bb9e209a2a1fd8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfe5126daefed30f2b750be6ec6fe6c5b836011ce2ae230840bb9e209a2a1fd8","first_computed_at":"2026-06-23T03:13:42.120559Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T03:13:42.120559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tJ/oKW9PFDQIX1KkwOXYQMoNQk2MwfCRMPZgCg9KLcYnQHxvnKE7mi1AOw5oYSqzDrLt3KSKzhOyQ3dzg+a4Ag==","signature_status":"signed_v1","signed_at":"2026-06-23T03:13:42.121203Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.09688","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ff63c2da64afe540d350f16230bec09879b252dfe724854063bd6958823cad7","sha256:f9192be7e62495ae3c48d555db5112fdfaa45a62bdd2a50c6f31ab68f67f4156"],"state_sha256":"0dba966435ae39784751a7cce858aedee7227e6f08725f5fafa6c3c54fc94151"}