FS, FP, PS, and PP stand for finite sums, finite products, pairwise sums and pairwise products respectively, all without repetition. Sometime shortly after the big bang it was shown that if a semigroup (S, ·) is finitely colored, there are a sequence ⟨x_n⟩_{1≤ n < ∞} and a color class which contains FP(⟨x_n⟩_{1≤ n < ∞}).

In 1999, Bergelson, Hindman, and Leader showed that if the real interval (0, 1) is finitely colored and each color class is measurable or each color class has the property of Baire, then there are a sequence ⟨x_n⟩_{1≤ n < ∞} and a color class which contains FS(⟨x_n⟩_{1≤ n < ∞})  FP(⟨x_n⟩_{1≤ n < ∞}).

The first of our results that I will discuss shows that there is a finite coloring of the positive reals such that each color class is either open or countable (so is measurable and has the property of Baire) and if  ⟨x_n⟩_{1≤ n < ∞}  is a sequence with PS(⟨x_n⟩_{1≤ n < ∞})  PP(⟨x_n⟩_{1≤ n < ∞}) monochromatic, then either x_n  approaches zero or x_n approaches infinity.

This presentation will be given via Zoom. The corresponding link will be sent to everyone in advance.