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This is a question of Hindman, Leader and Strauss.\n  Our aim in this short note is to show that the answer is negative. As an application, we disprove a conjectured equivalence between the two main forms of partition regularity, namely image partition regularity and kernel partition regularity."},"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":"1304.5173","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-04-18T16:04:49Z","cross_cats_sorted":[],"title_canon_sha256":"a9e60d94cdcac7d914e63f5356a05c6937c4f28c62fac535c7d1641dea4498eb","abstract_canon_sha256":"73c698a2a28183a93ba4b688d3478064193363b57a0b297d8699fa8b43430db5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:15.413557Z","signature_b64":"d0DYjdQ67pUpkRnn5iOlThPwaVwfSTSVlOIkRdlcC1um0qpyjMMeGyemWo5ETKh8C17glU6hJS8jSEXFMbOYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"239b13dd14ab47173e0b3b7870c8c05447881f62d293e1a0d5d2e13ea0c910be","last_reissued_at":"2026-05-18T03:04:15.412979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:15.412979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Partition regularity with congruence conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Barber, Imre Leader","submitted_at":"2013-04-18T16:04:49Z","abstract_excerpt":"An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. 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