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 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