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arXiv · 1401.2808

Ramsey Functions for Generalized Progressions

Mano Vikash Janardhanan·Sujith Vijay
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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.

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BibTeXRIS

Mano Vikash Janardhanan, Sujith Vijay. 2014-01-13. Ramsey Functions for Generalized Progressions. https://arxiv.org/abs/1401.2808

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