Abstrakt
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 inverted right perpendicular root N inverted left perpendicular 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 subset of 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 subset of P is kth-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 Omega(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.