# A New Lower Bound for van der Waerden Numbers

## TL;DR

This paper introduces a new recurrence relation for van der Waerden numbers, providing improved lower bounds and methods for constructing valid colorings, advancing understanding of these combinatorial quantities.

## Contribution

It establishes a novel recurrence relation for van der Waerden numbers, leading to better lower bounds and explicit coloring constructions.

## Key findings

- Proves a new recurrence relation for $w(r,k)$.
- Derives improved lower bounds for specific $r$ and $k$.
- Enables explicit construction of valid colorings.

## 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.