On Avoiding Arithmetic Progressions Whose Common Differences Belong to a Given Small Set

Abstract

It is well-known that if $D$ is any finite set of integers, then there is an $n$ large enough so that there exists a 2-coloring of the positive integers that avoids any monochromatic $n$-term arithmetic progressions whose common differences belong to $D$.If $\overrightarrow{d}=(d_1,\dots ,d_k)$ and $\overrightarrow{n}=(n_1,\dots ,n_k)$ are $k$-tuples of positive integers, denote by $f_{\overrightarrow{d}}(\overrightarrow{n})$ the least positive integer $N$, if it exists, such that for every 2-coloring of $[1,N]$ there is, for some $i$, a monochromatic $n_i$-term arithmetic progression with common difference $d_i$.This paper looks at the problem of determining when $f_{\overrightarrow{d}}(\overrightarrow{n})$ exists, and its value when it does exist, for $k\le 3$.A complete answer is given for $k=2$.A partial answer is given for $k=3$, including the fact that for all ordered triples $\overrightarrow{d}$, $f_{\overrightarrow{d}}(4,4,4)$ does not exist.