On a Question Raised by Brown, Graham and Landman
DOI:
https://doi.org/10.5644/SJM.01.1.03Keywords:
Arithmetic progressions, Van der Waerden's theoremAbstract
We construct non-periodic 2-colorings that avoid long mochromatic progressions having odd common differences. Also we prove that the set of all arithmetic progressions with common differences in $\left( \mathbb{N}\mathbf{!-}1\right) \cup \mathbb{N}!\cup \left( \mathbb{N}\mathbf{!+}1\right) - \{ 0\}$ does not have the $2$-Ramsey property.
2000 Mathematics Subject Classification. Primary 11B25; Secondary 05D10
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References
T. C. Brown, R. L. Graham and B. M. Landman, On the set of common differences in van der Waerden’s theorem on arithmetic progressions, Can. Math. Bull., 42 (1999), 25–36.
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, 1981.
B. M. Landman and A. Robertson, Avoiding monochromatic sequences with special gaps, Submitted.
B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskd., 15 (1927) 212–216.
