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 and 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–Čech compactification of the real numbers with the discrete topology. They are also obtained by elementary methods. In each case we in fact get significant strengthenings of the above stated results (with different strengthenings obtained by the algebraic and elementary methods). Some related (although weaker) results are established for arbitrary partitions of the rationals and the dyadic rationals, and a counterexample is given to show that even weak versions of the combined additive and multiplicative results do not hold in the dyadic rationals.