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We extend Frankl's theorem in a graph-theoretic direction. For a graph G, and r>=1, let P^r(G) be the family of all r-subsets of the vertex set of G such that every r-subset is either an independent set or contains a maximum independent set. We will consider k-wise intersecting subfamilies of this family for the graph "},"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":"1304.0242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-31T18:41:41Z","cross_cats_sorted":[],"title_canon_sha256":"2fc1c101b48d1d8224305521699d351cdaa8404589a7b16069f6d60f362685c1","abstract_canon_sha256":"f691ecb356ca1ad978009e66af9f1e3f20fe897cba96dac5b3ec3b7744dea7d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:17.569961Z","signature_b64":"CirFk8W00GBKEsj6FYrjnyDCjwWRL21wP9TCg1FnrPtxGI2Z2SjP0OKBHndncLw0mQ9Q7XX9tS0oL1oA4J8vAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"297dd409d74b6d206be36be95ecea8be3037671cfdd4213c70c5a501d502c650","last_reissued_at":"2026-05-18T03:29:17.569137Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:17.569137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On k-wise intersecting families of vertex sets in perfect matchings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vikram Kamat","submitted_at":"2013-03-31T18:41:41Z","abstract_excerpt":"We consider the following generalization of the seminal Erd\\H{o}s-Ko-Rado theorem, due to Frankl. 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