Ramsey Functions for Generalized Progressions

TL;DR

This paper establishes exponential lower bounds for the Ramsey function related to semi-progressions of fixed scope, advancing understanding of generalized progressions in combinatorial number theory.

Contribution

It provides the first exponential lower bounds for the Ramsey function of semi-progressions and improves bounds for quasi-progressions.

Findings

Exponential lower bounds for $S_m(k)$ when $m=O(1)$.

Marginal improvement on bounds for quasi-progressions.

Enhanced understanding of generalized progression structures in Ramsey theory.

Abstract

Given positive integers $n$ and $k$, a $k$-term semi-progression of scope $m$ is a sequence $(x_1,x_2,...,x_k)$ such that $x_{j+1} - x_j \in \{d,2d,\ldots,md\}, 1 \le j \le k-1$, for some positive integer $d$. Thus an arithmetic progression is a semi-progression of scope $1$. Let $S_m(k)$ denote the least integer for which every coloring of $\{1,2,...,S_m(k)\}$ yields a monochromatic $k$-term semi-progression of scope $m$. We obtain an exponential lower bound on $S_m(k)$ for all $m=O(1)$. Our approach also yields a marginal improvement on the best known lower bound for the analogous Ramsey function for quasi-progressions, which are sequences whose successive differences lie in a small interval.