Colorings with only rainbow arithmetic progressions

Abstract

If we want to color $1,2,\ldots ,n$ with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least n/2 colors. Surprisingly, much fewer colors suffice if we are allowed to leave a negligible proportion of integers uncolored. Specifically, we prove that there exist $\alpha ,\beta <1$ such that for every n, there is a subset A of $\{1,2,\ldots ,n\}$ of size at least $n-n^{\alpha }$, the elements of which can be colored with $n^{\beta }$ colors with the property that every 3-term arithmetic progression in A is rainbow. Moreover, $\beta$ can be chosen to be arbitrarily small. Our result can be easily extended to k-term arithmetic progressions for any $k\ge 3$.

As a corollary, we obtain a simple proof of the following result of Alon, Moitra, and Sudakov, which can be used to design efficient communication protocols over shared directional multi-channels. There exist $\alpha ',\beta '<2$ such that for every n, there is a graph with n vertices and at least $\left( {\begin{array}{c}n\\ 2\end{array}}\right) -n^{\alpha '}$ edges, whose edge set can be partitioned into at most $n^{\beta '}$induced matchings.