Some counterexamples in the partition calculus

We show that the pairs (2-element subsets; edges of the complete graph) of a set of cardinality א₁ can be colored with 4 colors so that every uncountable subset contains pairs of every color, and that the pairs of real numbers can be colored with א₀ colors so that every set of reals of cardinality 2^א0 contains pairs of every color. These results are counterexamples to certain transfinite analogs of Ramsey's theorem. Results of this kind were obtained previously by Sierpiński and by Erdös, Hajnal, and Rado. The Erdös-Hajnal-Rado result is much stronger than ours, but they used the continuum hypothesis and we do not. As by-products, we get an uncountable tournament with no uncountable transitive subtournament, and an uncountable partially ordered set such that every uncountable subset contains an infinite antichain and a chain isomorphic to the rationals. The tournament was pointed out to us by R. Laver, and is included with his permission.