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Let $c^\\exists_{f,g}$ be the dual notion: For every branch $\\nu$, one of the $g$-trees guesses $\\nu(m)$ infinitely often. We show that it is consistent that $c^\\exists_{f_\\epsilon,g_\\epsilon}=c^\\forall_{f_\\epsilon,g_\\epsilon}=\\kappa_\\epsilon$ for continuum many pairwise different cardinals $\\kappa_\\epsilon$ and suitable pairs $(f_\\epsilon,g_\\epsilon)$. For the proof"},"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":"1003.3425","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2010-03-17T18:28:37Z","cross_cats_sorted":[],"title_canon_sha256":"b13583ae96faf863ef3330d7e7c8a166003c0fc3805716157b0ca2bc75b7510a","abstract_canon_sha256":"c7750493855172eecdbc9c1d41e88b16fb6d96cb39c9a6f2c56e95a1567d1d11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:27.933511Z","signature_b64":"08E9NQ7DfVtAPrYEyUIQeN7XK54TAgChERDU8kfX8B2ecYckRzu9W0qMi49IyO4STXkQgryYR+Z7K9D0s+zpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35689a55b7df89df5dc637807a8cca724058fd8a38c48e5b832b6d2d69aa7184","last_reissued_at":"2026-05-18T04:05:27.933000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:27.933000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Creature forcing and large continuum: The joy of halving","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Jakob Kellner, Saharon Shelah","submitted_at":"2010-03-17T18:28:37Z","abstract_excerpt":"For $f,g\\in\\omega^\\omega$ let $c^\\forall_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e., for every branch $\\nu$ of the $f$-tree, one of the $g$-trees contains $\\nu$. Let $c^\\exists_{f,g}$ be the dual notion: For every branch $\\nu$, one of the $g$-trees guesses $\\nu(m)$ infinitely often. We show that it is consistent that $c^\\exists_{f_\\epsilon,g_\\epsilon}=c^\\forall_{f_\\epsilon,g_\\epsilon}=\\kappa_\\epsilon$ for continuum many pairwise different cardinals $\\kappa_\\epsilon$ and suitable pairs $(f_\\epsilon,g_\\epsilon)$. 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