Key points are not available for this paper at this time.
Hindman conjectured that any finite partition of N has a monochromatic \x, y, x+y, xy\. Recently, Bowen proved the result for all 2-partition. In this paper, we extend Bowen's result to any semiring (S, +, ) such that Ss is piecewise syndetic for all s S. As a method, we gave a combinatorial proof for a piecewise syndetic version of Bergerson and Glasscock's IPᵣ^* Szemer\'edi Theorem, and discussed the case when the operation is not commutative.
Tao et al. (Tue,) studied this question.