New York Journal of Mathematics
Volume 6 (2000) 73-86


Dennis Davenport, Neil Hindman, Imre Leader and Dona Strauss

Continuous Homomorphisms on βN and Ramsey Theory

Published: April 18, 2000
Keywords: continuous homomorphism, Ramsey Theory, Stone-Čech compactification
Subject: 05D10, 22A30

We consider the question of the existence of a nontrivial continuous homomorphism from (βN,+) into N*=βN\N. This problem is known to be equivalent to the existence of distinct p and q in N* satisfying the equations p+p=q=q+q=q+p=p+q. We obtain certain restrictions on possible values of p and q in these equations and show that the existence of such p and q implies the existence of p, q, and r satisfying the equations above and the additional equations r=r+r, p=p+r=r+p, and q=q+r=r+q. We show that the existence of solutions to these equations implies the existence of triples of subsets of N satisfying an unusual Ramsey Theoretic property. In particular, they imply the existence of a subset A with the property that whenever it is finitely colored, there is a sequence in the complement of A, all of whose sums two or more terms at a time are monochrome. Finally we show that there do exist sets satisfying finite approximations to this latter property.

Author information

Dennis Davenport:
Department of Mathematics, Miami University, Oxford, OH 45056, USA

Neil Hindman:
Department of Mathematics, Howard University, Washington, DC 20059, USA

Imre Leader:
Department of Mathematics, University College London, London WC1E6BT, UK

Dona Strauss:
Department of Pure Mathematics, University of Hull, Hull HU6 7RX, UK