Hindman's finite sums and products conjecture

Theorem 1.1: for every finite coloring of $\mathbb N$ and every $m\ge1$ there are $a_1<\dots<a_m$ such that every nonempty subset sum and subset product has one common color; the $a_d$ can be made to grow faster than any prescribed power of the earlier sums and products, so all $2(2^m-1)-m$ expressions are distinct. In particular $\{x,y,x+y,xy\}$ is monochromatic in every finite coloring of the positive integers. It gives no quantitative bounds and no density version, and the infinite version is false (Hindman 1980).

Theorem 1.1: for every finite coloring of $\mathbb N$ and every $m\ge1$ there are $a_1<\dots<a_m$ such that every nonempty subset sum and subset product has one common color; the $a_d$ can be made to grow faster than any prescribed power of the earlier sums and products, so all $2(2^m-1)-m$ expressions are distinct. In particular $\{x,y,x+y,xy\}$ is monochromatic in every finite coloring of the positive integers. It gives no quantitative bounds and no density version, and the infinite version is false (Hindman 1980).