  We establish sharpness for the threshold of van der Waerden's theorem in
random subsets of $\mathbb{Z}/n\mathbb{Z}$. More precisely, for $k\geq 3$ and
$Z\subseteq \mathbb{Z}/n\mathbb{Z}$ we say $Z$ has the van der Waerden property
if any two-colouring of $Z$ yields a monochromatic arithmetic progression of
length $k$. R\"odl and Ruci\'nski (1995) determined the threshold for this
property for any k and we show that this threshold is sharp.
  The proof is based on Friedgut's criteria (1999) for sharp thresholds, and on
the recently developed container method for independent sets in hypergraphs by
Balogh, Morris and Samotij (2015) and by Saxton and Thomason (2015).
