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Schilhan 2024 wetzel families continuum
corollary_4_5: Schilhan and Weinert's positive answer to Zapletal's Question 22: assuming GCH, arbitrarily large strongly almost disjoint families of functions in aleph_omega^aleph_(omega+1) can be added without collapsing cardinals.
lemma_3_2: Schilhan and Weinert's ZFC lemma: a Wetzel family has exactly 2^aleph_0 members, and for each lambda below the continuum, all but fewer than 2^aleph_0 complex numbers take at least lambda values under the family.
proposition_3_7: Schilhan and Weinert's ZFC proposition: if some set of fewer than 2^aleph_0 complex numbers is universal for entire functions, then a Wetzel family exists.
theorem_5_14: Schilhan and Weinert's main theorem: assuming GCH, for every infinite cardinal kappa of uncountable cofinality there is a cardinal and cofinality preserving forcing extension with 2^aleph_0 = kappa and a Wetzel family, in which Martin's Axiom also holds when kappa is regular.
theorem_6_5: Schilhan and Weinert's theorem that Martin's Axiom with the negation of CH implies that no universal set exists, and their corollary that the existence of a Wetzel family does not imply that of a universal set.
theorem_7_1: Schilhan and Weinert's theorem that over a model of CH some proper forcing extension preserving all cardinals and cofinalities has 2^aleph_0 = aleph_2 and the ground model's complex numbers form a universal set.
Schilhan, Jonathan and Weinert, Thilo, Wetzel families and the continuum. J. Lond. Math. Soc. (2) 109 (2024), no. 6, Paper No. e12918, 27, doi:10.1112/jlms.12918. The copy read for this card is arXiv:2310.19473v3 (stamped 13 May 2024), whose arXiv record names arXiv's non-exclusive distribution license, every other right reserved. The journal version's Crossref record (read 2026-10-07) names CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/); that version was not read.
A Wetzel family is a family F of entire functions that pointwise takes fewer than |F| values; Erdős showed the continuum hypothesis implies one exists and asked what happens without CH. Kumar and Shelah answered the existence question in the negative in the side-by-side Cohen model and produced a model with a Wetzel family of the singular cardinality aleph_(omega_1), leaving open whether a Wetzel family is consistent with the continuum equal to aleph_2. The main result here, Theorem 5.14, assumes GCH and takes any infinite cardinal kappa of uncountable cofinality; it gives a forcing extension, preserving cardinals and cofinalities, in which 2^(aleph_0) = kappa and a Wetzel family exists, and in which Martin's Axiom also holds when kappa is regular; this settles Kumar and Shelah's question and shows Wetzel families impose no restriction on the size of the continuum. Section 3 proves ZFC facts, including that every Wetzel family has cardinality 2^(aleph_0) (Lemma 3.2), Erdős's result that under CH countable dense sets are universal, and that a universal set for sets of complex numbers yields a Wetzel family (Proposition 3.7); Section 4 forces a family of strongly almost disjoint functions and in doing so answers Zapletal's Question 22 of the cited paper; Section 6 shows universal sets fail under Martin's Axiom plus not-CH, so the converse of Proposition 3.7 fails, while Section 7 forces a universal set with continuum aleph_2 using a proper forcing with pairs of models as side conditions. For Problem 1119, Theorem 5.14 with kappa = aleph_2 gives a model of 2^(aleph_0) = aleph_2 with a family of more than aleph_1 entire functions taking at most aleph_1 values at each point, a negative instance of the problem's question for m = aleph_1.
Source: https://arxiv.org/abs/2310.19473.
Bears on. #1119: the problem asks whether, for aleph_0 < m < 2^(aleph_0), every family of entire functions taking at most m values at each point has at most m members. Theorem 5.14 with kappa = aleph_2 gives a forcing extension of a GCH model in which 2^(aleph_0) = aleph_2 and such a family with aleph_2 members exists for m = aleph_1, so the answer is no there, in the case m^+ = 2^(aleph_0); with kappa = m^+ the same holds for every uncountable m. The paper states the kappa = aleph_2 case as the answer to Kumar and Shelah's question and does not address the case m^+ < 2^(aleph_0).
Results. Theorem 5.14 (p. 17), the main theorem; Lemma 3.2 (p. 5), with Definition 3.1; Proposition 3.7 (p. 6), with Definition 3.4 and Propositions 3.5 and 3.6; Theorem 6.5 and Corollary 6.6 (p. 21); Theorem 7.1 (p. 21); Corollary 4.5 (p. 10), with Proposition 4.1 (p. 7).
No file of this source is held: the arXiv copy read carries no license that permits its redistribution, the CC BY 4.0 journal version has not been fetched, and the card cites the edition it names above.