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In this paper Bergelson, Blass, and Hindman prove the following

Theorem 1.2 Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. Then $\bar s$ has a variable extraction $\bar t$ such that the set of extracted words of $\bar t$ is monochromatic.

I would like to know if there is an easy counter example to the claim that the components of $\bar t$ are obtained concatenating those of $\bar s$ in increasing order. (As stated, the theorem also allows substitution of the variable $v$ with elements of $\Sigma$ as long as all components of $\bar s$ contain at least a variable.)

Notation $W(\Sigma; v)$ is the set of words in the alphabeth $\Sigma\cup\{v\}$ that contain at least one occurrence of the variable $v$.

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If I understand the definitions correctly, you can take $\bar s$ to be a constant sequence equal to $v$. Take a coloring that colors words containing only the letter $a$ in red and words containing only the letter $b$ in blue. Then any $\bar t$ obtained by concatenating members of $\bar s$ will be composed of words in which only the letter $v$ is used. Therefore those words can be reduced to words using only $a$ or words using only $b$, which are of different colors. Hence $\bar t$ does not have the required property.

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  • $\begingroup$ Indeed a constant sequence produces a counter example. If I understand the definition correctly myself, you have to remain inside $W(\Sigma;v)$ i.e. words may not be constant. So the sequence such contain the word $vv$ and coloring should color words not containing the letter $b$ in red and everything else in blue. $\endgroup$
    – Domenico Zambella
    Commented Aug 17, 2018 at 7:53

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