Higher-dimensional Delta-systems

We investigate the effect of adding $\omega _2$ Cohen reals on graphs on $\omega _2$ , in particular we show that $\omega _2 \to (\omega _2, \omega : \omega )^2$ holds after forcing with $\mathsf {Add}(\omega , \omega _2)$ in a model of $\mathsf {CH}$ . We also prove that this result is in a certain sense optimal as $\mathsf {Add}(\omega , \omega _2)$ forces that $\omega _2 \not \to (\omega _2, \omega : \omega _1)^2$ .

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Polish Space Partition Principles and the Halpern–läuchli Theorem

LAMBIE-HANSON,

ZUCKER

2024

J. symb. log.

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The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern–Läuchli theorem, and the same forcing from Harrington’s proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.

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Higher-dimensional Delta-systems

Cited by 3 publications

References 15 publications

Alternatives to the Halpern-Läuchli theorem

Alternatives to the Halpern-Läuchli theorem

Complete Bipartite Partition Relations in Cohen Extensions

Polish Space Partition Principles and the Halpern–läuchli Theorem

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