Open Access
July, 2017 Forcing-theoretic aspects of Hindman's Theorem
J. Math. Soc. Japan 69(3): 1247-1280 (July, 2017). DOI: 10.2969/jmsj/06931247


We investigate the partial order $(\mathrm{FIN})^\omega$ of infinite block sequences, ordered by almost condensation, from the forcing-theoretic point of view. This order bears the same relationship to Hindman's Theorem as $\mathcal{P}(\omega) /\mathrm{fin}$ does to Ramsey's Theorem. While $(\mathcal{P}(\omega) / \mathrm{fin})^2$ completely embeds into $(\mathrm{FIN})^\omega$, we show this is consistently false for higher powers of $\mathcal{P} (\omega) / \mathrm{fin}$, by proving that the distributivity number $\mathfrak{h}_3$ of $(\mathcal{P} (\omega) /\mathrm{fin})^3$ may be strictly smaller than the distributivity number $\mathfrak{h}_{\mathrm{FIN}}$ of $(\mathrm{FIN})^\omega$. We also investigate infinite maximal antichains in $(\mathrm{FIN})^\omega$ and show that the least cardinality $\mathfrak{a}_{\mathrm{FIN}}$ of such a maximal antichain is at least the smallest size of a nonmeager set of reals. As a consequence, we obtain that $\mathfrak{a}_{\mathrm{FIN}}$ is consistently larger than $\mathfrak{a}$, the least cardinality of an infinite maximal antichain in $\mathcal{P} (\omega) / \mathrm{fin}$.


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Jörg BRENDLE. Luz María GARCÍA ÁVILA. "Forcing-theoretic aspects of Hindman's Theorem." J. Math. Soc. Japan 69 (3) 1247 - 1280, July, 2017.


Published: July, 2017
First available in Project Euclid: 12 July 2017

zbMATH: 06786997
MathSciNet: MR3685044
Digital Object Identifier: 10.2969/jmsj/06931247

Primary: 03E40
Secondary: 03E17 , 03E35

Keywords: almost disjointness number , Cohen forcing , complete embedding , diamond principle , distributivity number , Hechler forcing , Hindman's Theorem , Iterated forcing , Laver forcing , maximal almost disjoint family , random forcing

Rights: Copyright © 2017 Mathematical Society of Japan

Vol.69 • No. 3 • July, 2017
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