Higher-order Erdős–Szekeres theorems

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 Erdős–Szekeres theorem asserts that every such P contains a monotone subsequence S of ⌈ N ⌉ points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of Ω ( 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 ⊆ P to be positive if it lies on the graph of a function whose k th derivative is everywhere nonnegative, and similarly for a negative ( k + 1 ) -tuple. Then we say that S ⊆ P is k th-order monotone if its ( k + 1 ) -tuples are all positive or all negative. We investigate a quantitative bound for the corresponding Ramsey-type result (i.e., how large k th-order monotone subsequence can be guaranteed in every N -point  P ). We obtain an Ω ( log ( k − 1 ) N ) lower bound ( ( k − 1 ) -times iterated logarithm). This is based on a quantitative Ramsey-type theorem for transitive colorings of the complete ( k + 1 ) -uniform hypergraph (these were recently considered by Pach, Fox, Sudakov, and Suk). 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.