Aaron Robertson
Published 2012-06-13, updated 2014-07-03Version 2
We show that $\sqrt{k}\cdot r^{k/2}$ is a threshold interval length where, under mild conditions, almost every $r$-coloring of an interval of longer length contains a monochromatic $k$-term arithmetic progression, while almost no $r$-coloring of an interval of shorter length contains a monochromatic $k$-term arithmetic progression.
The number of subsets of integers with no $k$-term arithmetic progression
Integer colorings with no rainbow $k$-term arithmetic progression
Small subsets without $k$-term arithmetic progressions
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