Colorings with only rainbow arithmetic progressions
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Date
2020-08
Publication Type
Journal Article
ETH Bibliography
yes
OPEN ACCESS
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Abstract
If we want to color 1, 2, ... , n with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least n/2 colors. Surprisingly, much fewer colors suffice if we are allowed to leave a negligible proportion of integers uncolored. Specifically, we prove that there exist alpha, beta < 1 such that for every n, there is a subset A of {1, 2, ... , n} of size at least n - n(alpha), the elements of which can be colored with n(beta) colors with the property that every 3-term arithmetic progression in A is rainbow. Moreover, beta can be chosen to be arbitrarily small. Our result can be easily extended to k-term arithmetic progressions for any k >= 3. As a corollary, we obtain a simple proof of the following result of Alon, Moitra, and Sudakov, which can be used to design efficient communication protocols over shared directional multi-channels. There exist alpha', beta' < 2 such that for every n, there is a graph with n vertices and at least ((n)(2)) - n(alpha)' edges, whose edge set can be partitioned into at most n(beta)' induced matchings.
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published
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Journal / series
Volume
161 (2)
Pages / Article No.
507 - 515
Publisher
Springer
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Edition / version
Methods
Software
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Date collected
Date created
Subject
arithmetic progression; induced matching rainbow
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Notes
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.