Mathematics > Combinatorics
[Submitted on 4 Aug 2018 (v1), last revised 16 Jan 2020 (this version, v2)]
Title:Translation invariant filters and van der Waerden's Theorem
View PDFAbstract:We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise syndetically"-many monochromatic arithmetic progressions of any length k in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn's Lemma is needed.
Submission history
From: Mauro Di Nasso [view email][v1] Sat, 4 Aug 2018 16:02:12 UTC (9 KB)
[v2] Thu, 16 Jan 2020 09:36:55 UTC (9 KB)
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