Annals of Combinatorics 1 (1997) 245-252

On a Problem of Erdös and Rado

Jean A. Larson and William J. Mitchell

Department of Mathematics, University of Florida, Gainesville, FL 32611, USA
{jal; mitchell}

Received March 25, 1997

AMS Subject Classification: 05C55, 05C20, 03E10

Abstract. We give some improved estimates for the digraph Ramsey numbers r(K*n, Lm), the smallest number p such that any digraph of order p either has an independent set of n vertices or contains a transitive tournament of order m.
    By results of Baumgartner and of Erdös and Rado, this is equivalent to the following infinite partition problem: for an infinite cardinal k and positive integers n and m, find the smallest number p such that

k × p → (k × n, m)2,
that is, find the smallest number p so that any graph whose vertices are well ordered where order type k × p either has an independent subset of order type k × n or a complete subgraph of size m.

Keywords: digraph, Ramsey, partition, ordinal, graph, transitive tournament


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