New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions
Combinatorics
2012-01-20 v2
Abstract
Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-coloring of {1,2...,n} contains a monochromatic (a,b)-triple. Earlier results gave an upper bound on T(a,b;2) that is a fourth degree polynomial in b and a, and a quadratic lower bound. A new upper bound for T(a,b;2) is given that is a quadratic. Additionally, lower bounds are given for the case in which a = b, updated tables are provided, and open questions are presented.
Cite
@article{arxiv.1201.3842,
title = {New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions},
author = {Patrick Allen and Bruce M. Landman and Holly Meeks},
journal= {arXiv preprint arXiv:1201.3842},
year = {2012}
}
Comments
The newer version reflects an updated table of values of T(a,b)