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We reduce this conjecture to design-like conjectures, where the monochromatic components of the color classes are bicliques [X,Y] with nonempty blocks X and Y. We prove this conjecture for r<6.\n  We show that the width (the number of bicliques) in every color class of any spanning r-coloring is at most 2^{r-1} (and this i"},"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":"1212.6861","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-31T11:20:43Z","cross_cats_sorted":[],"title_canon_sha256":"da3a8cc23611bc3f1b4a900605ca29cca17fd36be3849f3cde3ab64f191976fa","abstract_canon_sha256":"a6f73ba3124e0a769fe3105271eb180b662f99ea1e906b1e082aa60ca1d448ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:30.678552Z","signature_b64":"GqK4dsw7x5fHdb7FOVa0UjGES3gCSeUG6Ic8+MdeMdV/v+Z+PkYyU6+euV3DXRUBmMQjLd+KS1y76N3ZhkrqCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25b7a7c9482feeb439691885cef81f6f1b24959b4d17d850dc55b93fcf97ace8","last_reissued_at":"2026-05-18T03:37:30.677633Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:30.677633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Around a biclique cover conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. 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