Let $W(3,k)$ be the van der Waerden number defined as the minimum $n$ such that in any red/blue colouring of $\{1,\dots ,n\}$ there exists either a red $3$-term arithmetic progression or a blue $k$-term arithmetic progression.Give reasonable bounds for $W(3,k)$. In particular, give any non-trivial lower bounds for $W(3,k)$ and prove that $W(3,k)<\exp (k^c)$ for some constant $c<1$.