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 x_1 < x_2 < ... < 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:   1. We show that for every fixed q we have N_3(q,n)=2^{\Theta(n^{q-1})}, thus resolving an open problem raised by Fox et al.   2. We show that for every k >= 3, N_k(2,n)=2^{\cdot^{\cdot^{2^{(2-o(1))n}}}} where the height of the tower is k-2, thus resolving an open problem raised by Elias and Matousek.   3. We give a new pigeonhole proof of the Erd\H{o}s-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdos-Szekeres Lemma on increasing/decreasing subsequences. 
Guy Moshkovitz and Asaf Shapira
math.CO