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arXiv · 1705.09673

A New Lower Bound for van der Waerden Numbers

Thomas Blankenship·Jay Cummings·Vladislav Taranchuk
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Abstract

In this paper we prove a new recurrence relation on the van der Waerden numbers, $w(r,k)$. In particular, if $p$ is a prime and $p\leq k$ then $w(r, k) > p \cdot \left(w\left(r - \left\lceil \frac{r}{p}\right\rceil, k\right) -1\right)$. This recurrence gives the lower bound $w(r, p+1) > p^{r-1}2^p$ when $r \leq p$, which generalizes Berlekamp's theorem on 2-colorings, and gives the best known bound for a large interval of $r$. The recurrence can also be used to construct explicit valid colorings, and it improves known lower bounds on small van der Waerden numbers.

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Thomas Blankenship, Jay Cummings, Vladislav Taranchuk. 2017-10-25. A New Lower Bound for van der Waerden Numbers. https://doi.org/10.1016/j.ejc.2017.10.007

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