 

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, StoneČech compactification 
Subject: 
05D10, 22A30 


Abstract
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
davenpde@casmail.muohio.edu
Neil Hindman:
Department of Mathematics, Howard University, Washington, DC 20059, USA
nhindman@howard.edu
http://members.aol.com/nhindman/
Imre Leader:
Department of Mathematics, University College London, London WC1E6BT, UK
i.leader@ucl.ac.uk
Dona Strauss:
Department of Pure Mathematics, University of Hull, Hull HU6 7RX, UK
d.strauss@maths.hull.ac.uk

