Milliken-Taylor system ($\mathrm{MT}(a,x)$)

For fixed positive integer coefficients $a_1,\dots ,a_m$ and a sequence of positive integers $x_1,x_2,\dots$, the Milliken-Taylor system isThe ordered blocks need not be consecutive intervals, but they are disjoint and every index of one block precedes every index of the next. This is a set of values from all allowed block choices, rather than just one weighted sum.

For $a=(1,2)$ and $x=(1,2,4,8,\dots )$, blocks $F_1=\{1\}$ and $F_2=\{2,3\}$ give $1+2(2+4)=13$. Choosing $F_1=\{1,3\}$ and $F_2=\{4\}$ gives $1+4+2(8)=21$.

The Milliken–Taylor theorem guarantees that for any finite coloring of the positive integers and fixed positive coefficient vector, some infinite generating sequence makes the entire system monochromatic. The special case $a=(1)$ is the finite-sums conclusion of Hindman's theorem.^[ref]