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.