Ramsey Theory, integer partitions and a new proof of the Erdos-Szekeres Theorem

Let H be a k-uniform hypergraph whose vertices are the integers 1, . . . , N. We say that H contains a monotone path of length n if there are _x1 < _x2 < ⋯ < _{x n +k -1} so that H contains all n edges of the form {_{x i}, _{x i +1}, . . . , _{x i +k -1}}. Let _{N k}(q, n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of _{N k}(q, n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdos and Szekeres, the problem of bounding _{N k}(q, n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.Our main contribution here is a novel approach for bounding the Ramsey-type numbers _{N k}(q, n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following:•We show that for every fixed q we have N3(q,n)=2Θ(nq-1), thus resolving an open problem raised by Fox et al.•We show that for every k ≥ 3, Nk(2,n)=2{dot operator}{dot operator}2(2-o(1))n where the height of the tower is k - 2, thus resolving an open problem raised by Eliáš and Matoušek.•We give a new pigeonhole proof of the Erdos-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdos-Szekeres Lemma on increasing/decreasing subsequences.