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Another note on intervals in the Hales-Jewett theorem

  • The Hales-Jewett Theorem states that any r-colouring of [m]ⁿ contains a monochromatic combinatorial line if n is large enough. Shelah's proof of the theorem implies that for m = 3 there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most r intervals. For odd r, Conlon and Kamčev constructed r–colourings for which it cannot be fewer than r intervals. However, we show that for even r and large n, any r–colouring of [3]ⁿ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most r−1 intervals. This is optimal and extends a result of Leader and Räty for r=2.

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Metadaten
Author:Nina Kamčev, Christoph Spiegel
Document Type:Article
Parent Title (English):Electronic Journal of Combinatorics
Volume:29
Issue:1
First Page:P1.62
Year of first publication:2022
DOI:https://doi.org/10.37236/6730
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