Lower Bounds on van der Waerden Numbers: Randomized- and Deterministic-Constructive
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Gasarch-2011-Lower bounds on van der.pdf
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Author(s)
Gasarch, William
Haeupler, Bernhard
Date Issued
March 2011
Journal
Electronic Journal of Combinatorics
Publisher
Electronic Journal of Combinatorics
Citation
Gasarch, William, and Bernhard Haeupler. "Lower Bounds on van der Waerden Numbers: Randomized- and Deterministic-Constructive." Electronic Journal of Combinatorics, Volume 18, Issue 1 (2011).
Version
Final published version
Abstract
The van der Waerden number W(k,2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W(k,2) are enormous. Much effort was put into developing lower bounds on W(k,2). Most of these lower bound proofs employ the probabilistic method often in combination with the Lovasz Local Lemma. While these proofs show the existence of a 2-coloring that has no monochromatic arithmetic progression of length k they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm.
This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on W(k,2) in this light. We show how known nonconstructive lower bound proofs based on the Lovasz Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.37236/551
