Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A short proof of a partition relation for triples. | A short proof of a partition relation for triples Albin L. Jones Department of Mathematics Kenyon College Gambier OH 43022 USA jones@ http jones Submitted September 3 1999 Accepted March 11 2000 Abstract We provide a much shorter proof of the following partition theorem of P. Erdos and R. Rado If X is an uncountable linear order into which neither 1 nor embeds then X a 4 3 for every ordinal a . We also provide two counterexamples to possible generalizations of this theorem one of which answers a question of E. C. Milner and K. Prikry. MR Subject Classifications 03E05 04A20 05A18 05D10 Keywords partition relations Ramsey theory real orders transfinite ordinal numbers triples 1 A brief introduction In 3 Theorem 31 pp. 447-457 P. Erdos and R. Rado proved the theorem cited in the abstract namely that if X is an uncountable linear order into which neither 1 nor embeds then X a 4 3 for every ordinal a . The proof they provided was quite complicated and difficult to follow. We thought it might be helpful to exhibit a simpler more elementary proof. 2 Notation and definitions We use standard set-theoretic notation as used in for example 4 5 and 7 . An order is a set X together with an ordering a binary relation on X which is transitive if x y and y z then x z and irreflexive never is x x . If the THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R24 2 ordering is trichotic if always either x y y x or x y then the order is a linear order. For any order X with ordering the inversion of X is the order X with underlying set X and ordering which is defined by putting x y if and only if y x. For example R R while is isomorphic to the negative integers with their usual ordering. It is traditional when defining and describing orders to omit explicit mention of their orderings whenever possible leaving them to be inferred from context or usage. We will continue this tradition as it greatly simplifies notation and seldom seems to leads to confusion. For any .
