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The Milliken–Taylor theorem gives monochromatic systems of separated block sums with a fixed compressed coefficient vector, such as $(1,2,3)$. Systems associated with two nonproportional compressed coefficient vectors need not have the same color: in particular, there is a finite coloring for which a <finite-sums set> and the $(1,2,3)$ Milliken–Taylor system cannot be monochromatic together.

Milliken–Taylor theorem

No. The incompatibility part of the <Milliken–Taylor theorem> provides a finite coloring of $\mathbb N$ for which the systems associated with the nonproportional compressed coefficient vectors $(1)$ and $(1,2,3)$ cannot be monochromatic in the same color. The first system is the <finite-sums set>
$$
\operatorname{FS}(x_i)=\left\{\sum_{i\in I}x_i:0<|I|<\infty\right\},
$$
while singleton separated blocks in the second include every $x_i+2x_j+3x_k$ with $i<j<k$. Thus a sequence satisfying the proposed union would contradict that finite coloring.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 130 / 3 / c / ii / Solution

{title2=$\operatorname{FS}(x_i)$}

For a sequence $(x_i)$, its finite-sums set is
$$
\operatorname{FS}(x_i)=\left\{\sum_{i\in F}x_i: \varnothing\ne F\subseteq\mathbb N\text{ finite}\right\}.
$$


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 Incoming links: Finite-sums set