Hindman theorem

Every finite coloring of the positive integers has an infinite sequence $(x_i)$ for which all nonempty finite sums of distinct terms have one color.

For a sequence $(x_i)$, its finite-sums set is

The Milliken–Taylor theorem gives monochromatic systems of separated block sums with a fixed compressed coefficient vector, such as $(1,2,3)$. Systems associated with two nonproportional compressed coefficient vectors need not have the same color: in particular, there is a finite coloring for which a finite-sums set and the $(1,2,3)$ Milliken–Taylor system cannot be monochromatic together.