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We describe a family of posets $\\mathcal{Q}$ and show that the following dichotomy holds: if $Q\\in\\mathcal{Q}$, then $\\textrm{FF}(w,Q) \\le 2^{c(\\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\\not\\in\\mathcal{Q}$, then $\\textrm{FF}(w,Q) \\ge 2^w - 1$."},"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":"1810.03807","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-09T04:25:07Z","cross_cats_sorted":[],"title_canon_sha256":"398c01fe868fa1f794a8020ca5e0c9e8ce9526098070aa3aff7debf947ad7372","abstract_canon_sha256":"0bcfa24cea669c55b2dc4a73efecd795255ffc94964bde36c172cc770a66066f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:44.480853Z","signature_b64":"HZYzgUj5ymytC+Uc4SRJJPYiBYw5Lvct25VDdKu+SxSK6sQkzolOpeh0lGIVUSqKQhmGOSR+c1QK54f/78L9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5ecd2adbd63e40d5960e5f4d2296eaf64455ab4b3405f10ee3a35026673ad83","last_reissued_at":"2026-05-18T00:03:44.480336Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:44.480336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Dichotomy Theorem for First-Fit Chain Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin G. 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