António Girão: Monotone Arrays And A Multidimensional Ramsey Theorem

 A foundational result in Ramsey theory appears in a paper of Erdős and Szekeres from 1935: any sequence of n^2 +1 distinct real numbers contains either an increasing or decreasing subsequence of length n+1$n+1$.   This simple result was one of the starting seeds for the development of Ramsey theory. We discuss a generalization of the Erdős-Szekeres theorem to monotone arrays. We will show how to improve a recent result proved by Buci\'c, Sudakov and Tran. More precisely, we will show that a doubly exponential upper bound holds in all dimensions. Finally, we will see how this is intimately connected to a generalisation of Ramsey Theorem on the cartesian product of cliques.