Ordered and partially-ordered variants of Ramsey's theorem
نویسندگان
چکیده
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii PRELIMINARIES AND NOTATION . . . . . . . . . . . . . . . . . . . . . ix CHAPTER 1. OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Graph Ramsey Number . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Arrow Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The Directed Ramsey Number . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Ordered Ramsey Numbers . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Applications of the Ordered Ramsey Number . . . . . . . . . . . . . . . 9 1.4.1 Erdős-Szekeres Type Problems . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Track Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Partially-Ordered Ramsey Numbers . . . . . . . . . . . . . . . . . . . . . 12 CHAPTER 2. ORDERED RAMSEY NUMBERS OF HYPERGRAPHS 13 2.1 Ordered Ramsey Numbers of Loose Paths . . . . . . . . . . . . . . . . . 16 2.1.1 A “Stepping Up” Argument for Loose Paths . . . . . . . . . . . . 16
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