Let P = (p1, p2,..., p N) be a sequence of points in the plane, where p i = (x i, y i) and x1 < x2 < ⋯ < x N. A famous 1935 Erdos-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 kth 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 kth-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 R3, as well as for a Ramsey-type theorem for hyperplanes in R4 recently used by Dujmović and Langerman. © 2013 Elsevier Ltd.
Eliáš, M., & Matoušek, J. (2013). Higher-order Erdos-Szekeres theorems. Advances in Mathematics, 244, 1–15. https://doi.org/10.1016/j.aim.2013.04.020