Three-color van der Waerden Numbers Grow Super-exponentially

The van der Waerden number w(k;r) is the minimum positive integer N such that every r-coloring of the positive integers up to N contains a monochromatic k-term arithmetic progression. Estimating these numbers has remained a challenging open problem for the past century. In this talk, we will sketch a proof that the three-color van der Waerden number w(k;3) grows faster than any exponential in k. The proof uses a novel probabilistic construction. This settles several longstanding conjectures in the area.

Based on joint work with Zach Hunter.