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Assume $K^T$ has joint embedding and amalgamation.\n  If $\\lambda^{|T|}=\\lambda$ or $\\forall \\mu < \\lambda( \\mu^{|T|} < \\lambda)$, then $K^T$ has a universal model of cardinality $\\lambda$.\n  As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the existence of universal reduced torsion-free"},"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":"1903.00414","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-03-01T17:12:09Z","cross_cats_sorted":["math.AC","math.RA"],"title_canon_sha256":"396e686b45d8b7584200d5767655963121b9d54b92b5009a0fa13a5a665243ad","abstract_canon_sha256":"758879858d1dfb77c120066cba73f7e79bfbd8f07b95f70fbb0fb578d6194e1f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:42:51.971468Z","signature_b64":"omifmDUTEV/EM0k87/rb470nCXQ/BalZxquIlQUgEr1BamfZP2XAroQVvGiLvIs6XMf3bPQ/LoLcW+dWaibWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0baaa7e8f94f1a8257fe37fb05153a114261e6b79e1dd9fbbece7531d4436dda","last_reissued_at":"2026-07-05T00:42:51.970912Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:42:51.970912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On universal modules with pure embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.RA"],"primary_cat":"math.LO","authors_text":"Marcos Mazari-Armida, Thomas G. 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