Bounds on some van der Waerden numbers |
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Authors: | Tom Brown Aaron Robertson |
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Affiliation: | a Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada b Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA c Department of Mathematics, Colgate University, Hamilton, NY 13346, USA |
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Abstract: | For positive integers s and k1,k2,…,ks, the van der Waerden number w(k1,k2,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided. |
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Keywords: | Ramsey theory van der Waerden numbers Arithmetic progressions |
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