Additive and multiplicative Ramsey theory in the reals and the rationals

Let a finite partition F of the real interval (0, 1) be given. We show that if every member of F is measurable or if every member of F is a Baire set, then one member of F must contain a sequence with all of its finite sums a nd products (and, in the measurable case, all of its infinite sums as well) . These results are obtained by using the algebraic structure of the Stone- Cech compactification of the real numbers with the discrete topology. They are also obtained by elementary methods. In each case we in fact get signif icant strengthenings of the above stated results (with different strengthen ings obtained by the algebraic and elementary methods). Some related (altho ugh weaker) results are established for arbitrary partitions of the rationa ls and the dyadic rationals, and a counterexample is given to show that eve n weak versions of the combined additive and multiplicative results do nut hold in the dyadic rationals. (C) 1999 Academic Press.