{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1994:HX7DOPVY5XMVLCBYCQJD24XKLZ","short_pith_number":"pith:HX7DOPVY","schema_version":"1.0","canonical_sha256":"3dfe373eb8edd955883814123d72ea5e46c6de23e2effaa57276d9db9868d073","source":{"kind":"arxiv","id":"math/9409214","version":1},"attestation_state":"computed","paper":{"title":"Invertible families of sets of bounded degree","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emanuel Knill","submitted_at":"1994-09-16T00:00:00Z","abstract_excerpt":"Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H is not invertible but every hypergraph obtained by removing an edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =< (d-1) {2d-1 choose d} + 1. The proof of this result leads to the f"},"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":"math/9409214","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"1994-09-16T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"cdc907f23ccdd4e54cb9fe6e0819718d6d15a44ab337ec56a622a76fedc79e46","abstract_canon_sha256":"382e4e97c38a470728bface3d63655f89b1318dca43e7d6a4e40bb4e693faff7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.103846Z","signature_b64":"5Q1a1pa/ISLx8WfRepc1G5AGJDiolVgoj7inB6WVbTMQAn6VQlk0KQEv9NTJ8Lh+LXgi8rD10V02L9Fa6K7GDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3dfe373eb8edd955883814123d72ea5e46c6de23e2effaa57276d9db9868d073","last_reissued_at":"2026-05-18T01:05:51.103373Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.103373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invertible families of sets of bounded degree","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emanuel Knill","submitted_at":"1994-09-16T00:00:00Z","abstract_excerpt":"Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H is not invertible but every hypergraph obtained by removing an edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =< (d-1) {2d-1 choose d} + 1. 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