Let P = (p1, p2, . . . , pN ) be a sequence of points in the plane, where pi = (xi, yi) and x1 < x2 < · · · < xN . 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. 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 kth-order monotone if its (k + 1)- tuples are all positive or all negative. In this thesis we investigate quantitative bound for the corresponding Ramsey-type result. We obtain an Ω(log(k−1) N) lower bound ((k − 1)-times iterated logarithm). We also improve bounds for related problems: Order types and One-sided sets of hyperplanes. 1
| Identifer | oai:union.ndltd.org:nusl.cz/oai:invenio.nusl.cz:304489 |
| Date | January 2012 |
| Creators | Eliáš, Marek |
| Contributors | Matoušek, Jiří, Cibulka, Josef |
| Source Sets | Czech ETDs |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/masterThesis |
| Rights | info:eu-repo/semantics/restrictedAccess |
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