Jacob Fox of Stanford University presents a Discrete Mathematics seminar at the Institute for Advanced Study on the van der Waerden number w(k;3), the smallest N such that every three-coloring of the integers up to N must contain a monochromatic k-term arithmetic progression. Fox sketches a proof, joint work with Zach Hunter, that this number grows faster than any exponential function of k, resolving several conjectures that have stood for decades about how fast these numbers can grow. He walks through the probabilistic construction behind the lower bound, situating the result against a century of partial progress on van der Waerden's theorem and related Ramsey-type problems. The talk is aimed at a seminar audience already familiar with Ramsey theory and arithmetic progressions, and it moves at the pace of a working research presentation rather than an introductory course, building up the combinatorial machinery piece by piece before arriving at the main growth-rate argument.