Abstract

For $D\subseteq {\mathbb{Z}}^{+}$, a sequence $x_1<x_2<\cdots <x_k$ is called a $D$-diffsequence if $x_i-x_{i-1}\in D$ for all $i\in \{2,\dots <!--FUNCTION\; APPLICATION-->,k\}$. Given a positive integer $r$, we say that $D\subseteq {\mathbb{Z}}^{+}$ is $r$-accessible if every $r$-coloring of ${\mathbb{Z}}^{+}$ admits arbitrarily long monochromatic $D$-diffsequences. If every $r$-coloring of ${\mathbb{Z}}^{+}$ admits arbitrarily long monochromatic arithmetic progressions $x$, $x+d$, $x+2d,\dots <!--FUNCTION\; APPLICATION-->,x+(k-1)d$, with $d\in D$, we say that $D$ is $r$-large. We prove some new results on accessibility. We also define a property that is stronger than accessibility but weaker than largeness, and study the connections among these various properties. The paper also includes bounds, as well as tables of exact values, for the associated Ramsey functions for some specific choices of $D$.

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