Some New Exact van der Waerden Numbers

Abstract

For positive integers $r,k_0,k_1,...,k_{r-1},$ the van der Waerden number $w(k_0,k_1,...,k_{r-1})$ is the least positive integer $n$ such that whenever $\{1,2,...,n\}$ is partitioned into $r$ sets $S_0,S_1,...,S_{r-1}$, there is some $i$ so that $S_i$ contains a $k_i$-term arithmetic progression. We find several new exact values of $w(k_0,k_1,...,k_{r-1})$. In addition, for the situation in which only one value of $k_i$ differs from 2, we give a precise formula for the van der Waerden function (provided this one value of $k_i$ is not too small)

11 pages

Cited by in corpus (8)

• On the van der Waerden numbers w(2;3,t)
• Exact Ramsey Theory: Green-Tao numbers and SAT
• New Lower Bounds for van der Waerden Numbers Using Distributed Computing
• Enumeration of three term arithmetic progressions in fixed density sets
• Bounds on Van der Waerden Numbers and Some Related Functions
• The Asymptotic Behaviors of ${\log }_{r}W(r,k)$ and ${\log }_{k}W(r,k)$, when $W(r,k)$ is a van der Waerden Number
• How to find the least upper bound on the van der Waerden Number $W(r,k)$ that is some integer Power of the coloring Integer $r$
• A simple Proof that $r^{k^2}$ is an upper Bound on each van der Waerden Number $W(r,k)$, for each $k$ bounded below by a certain positive real Number $\epsilon$