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We study, $\\chi(H)$, the classical chromatic number, and the $(2,2)$-spectrum of $H$, that is, the set of integers $k$ for which $H$ has a $(2,2)$-colouring using exactly $k$ colours.\n  We"},"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":"1402.3057","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-13T08:20:04Z","cross_cats_sorted":[],"title_canon_sha256":"3ace426399b60153b03e6ca919187d98eeb167c07e238e3bf4308e35c8fbd429","abstract_canon_sha256":"9d093f0fb535228f219e7a2c8dd1c3660359685edb9710f4028a83be1118b663"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:07.920067Z","signature_b64":"7Cklt/3LUqPvrO0MUfLqzma6MRRU1MxgjwyeUQHHxLnMUnSz5cs1qlYphaoR00UxuAmJ2YlYw7ScE52nXz4gBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e0caa0cd4263dacc06f0399bd2e6e92dbe7794ec75908ddc29b32298a673b2b","last_reissued_at":"2026-05-18T02:59:07.919302Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:07.919302Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$(2,2)$-colourings and clique-free $\\sigma$-hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Christina Zarb, Josef Lauri, Yair Caro","submitted_at":"2014-02-13T08:20:04Z","abstract_excerpt":"We consider vertex colourings of $r$-uniform hypergraphs $H$ in the classical sense, that is such that no edge has all its vertices given the same colour, and $(2,2)$-colourings of $H$ in which the vertices in any edge are given exactly two colours. 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