New lower bounds for van der Waerden numbers

Colour ${1,\ldots,N}$ red and blue, in such a manner that no 3 of the blue elements are in arithmetic progression. How long an arithmetic progression of red elements must there be? It had been speculated based on numerical evidence that there must always be a red progression of length about $\sqrt{N}$. I will describe a construction which shows that this is not the case - in fact, there is a colouring with no red progression of length more than about $\exp{\left(\left(\log{N}\right)^{3/4}\right)}$, and in particular less than any fixed power of $N$.

I will give a general overview of this kind of problem (which can be formulated in terms of finding lower bounds for so-called van der Waerden numbers), and an overview of the construction and some of the ingredients which enter into the proof. The collection of techniques brought to bear on the problem is quite extensive and includes tools from diophantine approximation, additive number theory and, at one point, random matrix theory