Continuous Homomorphisms on βN and Ramsey Theory

摘要

We consider the question of the existence of anontrivial continuous homomorphism from (βN,+)into N*=βNN. This problemis known to be equivalent to the existence of distinctp and q in N* satisfying the equationsp+p=q=q+q=q+p=p+q. We obtain certain restrictionson possible values of p and q in these equations andshow that the existence of such p and q impliesthe existence of p, q, and r satisfyingthe equations above and the additional equationsr=r+r, p=p+r=r+p, and q=q+r=r+q. Weshow that the existence of solutions to these equationsimplies the existence of triples of subsets of Nsatisfying an unusual Ramsey Theoretic property. In particular,they imply the existence of a subset A with the propertythat whenever it is finitely colored, there is a sequencein the complement of A, all of whose sums two ormore terms at a time are monochrome.Finally we show that there do exist sets satisfying finite approximationsto this latter property.