A conjecture at the heart of Ramsey theory, and surrounding questions on tensors and polynomials.

Seminar in Mathematics
This talk will primarily involve two phases. In the first half, we will focus on describing how the polynomial density Hales-Jewett conjecture, a central open problem in combinatorics, motivates a succession of three probabilistic and analytic results, where the proof of each result uses the next result as a black box: an approximation theorem on sets defined by polynomial conditions, a generalisation of the theorem of Green and Tao on the equidistribution of polynomials, and the existence of high-rank subtensors of high-rank tensors. In the second half, we will present additional results within the two nascent areas encompassing the previous results: the basic properties of the ranks of tensors, and the structure of objects defined in terms of linear forms and polynomials restricted to power sets such as the Boolean cube.