= Milliken–Taylor theorem
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Fix <positive integers> $a_1,\ldots,a_m$. A <Milliken-Taylor system> generated by a <sequence> $x_1,x_2,\ldots$ of <positive integers> is
$$
\operatorname{MT}(a,x)=\left\{\sum_{j=1}^ma_j\sum_{t\in F_j}x_t:\ F_j\subseteq\mathbb N\text{ finite and nonempty},\ \max F_j<\min F_{j+1}\right\}.
$$
The <Milliken–Taylor theorem> says that every <finite colouring> of the <positive integers> admits an infinite <sequence> $x$ for which this whole <set> is <monochromatic>. The block separation prevents the same index from contributing at two different <coefficient> positions. For $(a_1,a_2,a_3)=(1,2,3)$, choices $F_1=\{1,3\}$, $F_2=\{4\}$, $F_3=\{6,9\}$ give $x_1+x_3+2x_4+3x_6+3x_9$. For $m=1$, $a_1=1$, the statement is the <Hindman theorem>. One can also require $x$ to consist of successive disjoint block sums from any prescribed positive <sequence>.<Polynomial Extensions of the Milliken-Taylor Theorem by Vitaly Bergelson and others>[Definition1.5 and Theorem1.6]{ref}
A <compressed coefficient vector> has no adjacent repeated entries. Two such positive <coefficient> <vectors> can be forced to produce <Milliken-Taylor systems> in the same <colour class> for every <finite colouring> precisely when one is a positive rational multiple of the other. Thus a <finite colouring> can prevent a <finite-sums set> and a $(1,2,3)$ <Milliken-Taylor system> from sharing a colour, even with different generating <sequences>.<Separating Milliken-Taylor Systems with Negative Entries by Neil Hindman and others>[Definition1.5 and Theorem1.7]{ref}
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