Higher-order Erdõs-Szekeres theorems

Marek Eliáš

Abstract

Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos-Szekeres theorem asserts that every such P contains a monotone subsequence S of sqrt(N) points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of Omega(log N) points.

Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a (k+1)-tuple K\subseteq P to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative (k+1)-tuple. Then we say that S\subseteq P is kth-order monotone if its (k+1)-tuples are all positive or all negative.

We investigate quantitative bound for the corresponding Ramsey-type result (i.e., how large kth-order monotone subsequence can be guaranteed in every N-point P). We obtain an Omega(log^{(k-1)}N) lower bound ((k-1)-times iterated logarithm). This is based on a quantitative Ramsey-type theorem for what we call transitive colorings of the complete (k+1)-uniform hypergraph; it also provides a unified view of the two classical Erdos--Szekeres results mentioned above.

For k=3, we construct a geometric example providing an O(log log N) upper bound, tight up to a multiplicative constant. As a consequence, we obtain similar upper bounds for a Ramsey-type theorem for order-type homogeneous subsets in R^3, as well as for a Ramsey-type theorem for hyperplanes in R^4 recently used by Dujmovic and Langerman.