New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions

TL;DR

This paper establishes new quadratic upper bounds and improved lower bounds for van der Waerden-type numbers related to generalized 3-term arithmetic progressions, advancing understanding of their combinatorial properties.

Contribution

The paper introduces a quadratic upper bound for T(a,b;2) and provides updated lower bounds and tables for the case a=b, enhancing previous polynomial bounds.

Findings

New quadratic upper bounds for T(a,b;2)

Improved lower bounds for the case a=b

Updated tables and open questions

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.