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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On the Erdős–Dushnik–Miller theorem without AC

Banerjee Amitayu, Gopaulsingh Alexa

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 1

1--21

In ZFA (Zermelo–Fraenkel set theory with the Axiom of Extensionality weakened to allow the existence of atoms), we prove that the strength of the proposition EDM (“If G = (VG,EG) is a graph such that VG is uncountable, then for every coloring f : [VG]2 → {0, 1} either there is an uncountable set monochromatic in color 0, or there is a countably infinite set monochromatic in color 1”) is strictly between DCN1 (where DCN1 is Dependent Choices for N1, a weak choice form stronger than Dependent Choices (DC)) and Kurepa’s principle (“Any partially ordered set such that all of its antichains are finite and all of its chains are countable is countable”). Among other new results, we study the relations of EDM to BPI (Boolean Prime Ideal Theorem), RT (Ramsey’s theorem), De Bruijn–Erdős’ theorem for n-colorings, König’s lemma and several other weak choice forms. Moreover, we answer a part of a question raised by Lajos Soukup.

On the existence of almost disjoint and MAD families without AC

Tachtsis Eleftherios

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 2

101--124

In set theory without the Axiom of Choice (AC), we investigate the deductive strength and mutual relationships of the following statements: 1) Every infinite set X has an almost disjoint family A of infinite subsets of X with [formula]. (2) Every infinite set X has an almost disjoint family A of infinite subsets of X with [formula]. (3) For every infinite set X, every almost disjoint family in X can be extended to a maximal almost disjoint family in X. (4) For every infinite set X, no infinite maximal almost disjoint family in X has cardinality [formula]. (5) For every infinite set A, there is a continuum sized almost disjoint family A ⊆ Aω. (6) For every free ultrafilter U on ω and every infinite set A, the ultrapower Aω/U has cardinality at least [formula].