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Published: 
April 18, 2000

Keywords: 
continuous homomorphism, Ramsey Theory, StoneCech compactification

Subject: 
05D10, 22A30

Abstract:

We consider the question of the existence of a
nontrivial continuous homomorphism from $(\beta N,+)$
into $ N^{*}=\beta N\backslash 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 {\it 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
 