Accurate Bounds for New Van Der Waerden Type Numbers

Abstract

Numbers similar to those of van der Waerden are examined by considering sequences of positive integers $\{x_1,x_2,\dots ,x_n\}$ with $x_{i+1}=x_i+d+r_i$, where $d\in Z^{+}$ and $0\le r_i\le max(0,f(i))$ for a given function $f$ defined on $Z^{+}$. Let $w_f(n)$ denote the least positive integer such that if $\{1,2,\dots ,w_f(n)\}$ is $2$-colored, then there exists a monochromatic sequence of the type just described. Tables are given of $w_f(n)$ where $f(i)=i–k$ for various constants $k$, and also where $f(i)=i$ if $i\ge 2$, $f(1)=0$. In this latter case, as well as for $f(i)=i$, an upper bound is given that is very close to the actual values. A tight lower bound and fairly reasonable upper bound are given in the case $f(i)=i–1$.