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Van der Waerden spaces and Hindman spaces are not the same

Menachem Kojman·Saharon Shelah
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Abstract

A Hausdorff topological space X is van der Waerden if for every sequence (x_n)_n in X there is a converging subsequence (x_n)_{n in A} where subset A of omega contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (x_n)_n in X there is an IP-converging subsequence (x_n)_{n in FS(B)} for some infinite subset B of omega. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

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Menachem Kojman, Saharon Shelah. 2001-12-23. Van der Waerden spaces and Hindman spaces are not the same. https://arxiv.org/abs/math/0112265

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