= Milliken-Taylor system
{c}
{title2=$\operatorname{MT}(a,x)$}
For fixed positive integer coefficients $a_1,\ldots,a_m$ and a sequence of positive integers $x_1,x_2,\ldots$, the Milliken-Taylor system is
$$
\operatorname{MT}(a,x)=\left\{\sum_{j=1}^ma_j\sum_{t\in F_j}x_t:\ F_j\text{ finite and nonempty},\ \max F_j<\min F_{j+1}\right\}.
$$
The 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 $x=(1,2,4,8,\ldots)$, 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>.<Polynomial Extensions of the Milliken-Taylor Theorem by Vitaly Bergelson and others>[Definition1.5 and Theorem1.6]{ref}
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