  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.
