Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem

Abstract

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 Erdős 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 N 3 (q, n) = 2 Θ(n q−1 ) , thus resolving an open problem raised by Fox et al.• We show that for every k ≥ 3, N k (2, n) = 2where 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 Erdős-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdős-Szekeres Lemma on increasing/decreasing subsequences.

Ordered Ramsey theory and track representations of graphs

“…It is a general form of the Erdős-Szekeres Theorem, as observed by Seidenberg [20] and by Moshkovitz and Shapira [18]. Its proof provides motivation for the general argument.…”

“…Indeed, the general case is no more difficult. Thus it is proper to view it as due to Moshkovitz and Shapira [18]. Indeed, the proof we give here uses essentially the same ideas as their proof.…”

“…The result is easier to state and prove directly in terms of down-sets as given above. The inductive nature of J k−1 (Q) yields recurrences for upper and lower bounds on OR t (P k r ) as immediate corollaries, which in turn yield the bounds in [18].…”

“…Eliáš and Matoušek [6] asked more generally for the growth rate of N k (t, m). Moshkovitz and Shapira [18] answered with upper and lower bounds that are towers of the same height. For a nonnegative integer h, define tow h (x) by tow h (x) = 2…”

“…In our notation, their bounds are as follows: Theorem 1.2 (Moshkovitz and Shapira [18]). For t ≥ 2, k ≥ 3, and m = r − k + 1 ≥ 2,…”