Ramsey's theorem for sums, products, and arithmetic progressions (Q1059631)

Using ultrafilter techniques this work combines the classical result of van der Waerden's on arithmetic progressions and the well-known theorem of Hindman on finite sums from sequences within cells of a partition of N. Theorem. Let \({\mathbb{N}}=A_ 1\cup...\cup A_ r\). Then some \(A_ i\) contains simultaneously i) arbitrarily long arithmetic progressions, ii) an infinite set \(\{x_ i\}_{i\in \omega}\) with all its finite sums \(\sum_{I}x_ i\), iii) an infinite set \(\{y_ i\}_{i\in \omega}\) with all its finite products \(\prod_{I}x_ i\) (I\(\subseteq {\mathbb{N}}\), finite, nonempty).