Kumar 2017 question about families entire functions

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theorem_2_1: Kumar and Shelah's theorem that if c = lambda >= cf(lambda) > kappa = omega_1 and kappa Cohen reals are added, then in the extension every family of continuum many pairwise distinct entire functions takes continuum many values at some complex number.

theorem_3_1: Kumar and Shelah's theorem that it is consistent with ZFC plus the negation of CH that some family of continuum many entire functions takes fewer than continuum many values at every complex number.

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Kumar, Ashutosh and Shelah, Saharon, On a question about families of entire functions. Fund. Math. 239 (2017), no. 3, 279--288, doi:10.4064/fm252-3-2017. The copy read for this card is the Shelah archive's preprint (Sh:1078, version of 2017-01-05), which prints no copyright or license line on pp. 1--2 or 11--12; the archive's paper page (https://shelah.logic.at/papers/1078/, read 2026-10-02) states no terms, and the archive's legal notice (https://shelah.logic.at/impressum/, read 2026-10-02) states "Some documents on the site are copyrighted, and provided for 'fair use' in research. We do not own (and thus do not and cannot transfer or grant) any copyright to these documents.", every other right reserved.

Erdos asked whether there is a family F of entire functions with |F| = continuum such that for each z in C the set {f(z) : f in F} has size less than the continuum (Question 1.1, p. 1); the authors prove this question is undecidable in ZFC together with the negation of the continuum hypothesis. Theorem 2.1 (p. 2) shows the answer is no in Cohen extensions: if c = lambda >= cf(lambda) > kappa = omega_1 and one adds kappa Cohen reals, then for every family of c pairwise distinct entire functions some z has |{f(z) : f in F}| = c, the proof using that a Cohen-generic z avoids all meager sets coded earlier and that distinct entire functions agree on only a countable set. Theorem 3.1 (p. 2) gives the consistency of a yes answer with the failure of CH; its printed statement indexes the value set by z in C, a misprint for f in F. It is built by a finite-support iteration of ccc forcings of length omega_1 over a model of c = omega_{omega_1} (Lemma 3.2, p. 3) that adds, at stage i, a family of size omega_{i+1} whose values on the first omega_{j+1} complex numbers have size at most omega_{j+1} for each j <= i (Lemma 3.4, p. 5); this exploits the singularity of the continuum and adapts Erdos's CH construction, in which each function sends a countable set to rational complex numbers. At each point the value set of the resulting family is bounded below c, but not uniformly in the point. Question 4.1 (p. 12) asks whether a yes answer to Question 1.1 is consistent with 2^aleph_0 = aleph_2, and the paper leaves it open. Labels and pages are those of the preprint named above; the journal's numbering was not compared.

Source: https://shelah.logic.at/papers/1078/.

Read status. Claims checked: Theorems 2.1 and 3.1 and Questions 1.1 and 4.1 were read clause by clause on the printed pages, with the proof of Theorem 2.1. The forcing constructions for Theorem 3.1 (pp. 3--11) were read but not checked step by step.

Bears on. #1119: Theorem 2.1 with lambda = aleph_2 gives a model with c = aleph_2 in which every family of c pairwise distinct entire functions takes c values at some point, so the problem's question has answer yes for m = aleph_1, the case m^+ = c. Theorem 3.1 bounds the value sets below c but not by one m, and the paper does not relate it to a fixed m; its Question 4.1, left open there, asks for the model with c = aleph_2 that would give the answer no for m = aleph_1.

Results.

• Theorem 2.1 (p. 2): if c = lambda >= cf(lambda) > kappa = omega_1 and kappa Cohen reals are added, every family of c pairwise distinct entire functions takes c values at some point of C.
• Theorem 3.1 (p. 2): it is consistent with ZFC plus not-CH that some family of c entire functions takes fewer than c values at every point of C; the page also records Question 4.1 (p. 12).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.