A Probabilistic Threshold for Monochromatic Arithmetic Progressions
  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. 
Aaron Robertson
math.CO