Ramsey numbers and monotone colorings

Abstract

For positive integers N and r≥2, an r-monotone coloring of ({1,…,N}r) is a 2-coloring by −1 and +1 that is monotone on the lexicographically ordered sequence of r-tuples of every (r+1)-tuple from ({1,…,N}r+1). Let R‾mon(n;r) be the minimum N such that every r-monotone coloring of ({1,…,N}r) contains a monochromatic copy of ({1,…,n}r).For every r≥3, it is known that R‾mon(n;r)≤towr−1(O(n)), where towh(x) is the tower function of height h−1 defined as tow1(x)=x and towh(x)=2towh−1(x) for h≥2. The Erdős–Szekeres Lemma and the Erdős–Szekeres Theorem imply R‾mon(n;2)=(n−1)2+1 and R‾mon(n;3)=(2n−4n−2)+1, respectively. It follows from a result of Eliáš and Matoušek that R‾mon(n;4)≥tow3(Ω(n)).We show that R‾mon(n;r)≥towr−1(Ω(n)) for every r≥3. This, in particular, solves an open problem posed by Eliáš and Matoušek and by Moshkovitz and Shapira. Using two geometric interpretations of monotone colorings, we show connections between estimating R‾mon(n;r) and two Ramsey-type problems that have been recently considered by several researchers. Namely, we show connections with higher-order Erdős–Szekeres theorems and with Ramsey-type problems for order-type homogeneous sequences of points.We also prove that the number of r-monotone colorings of ({1,…,N}r) is 2Nr−1/rΘ(r) for N≥r≥3, which generalizes the well-known fact that the number of simple arrangements of N pseudolines is 2Θ(N2).