Milliken–Taylor theorem

Fix positive integers $a_1,\dots ,a_m$. A Milliken-Taylor system generated by a sequence $x_1,x_2,\dots$ of positive integers isThe 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.^[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.^[ref]

A compressed coefficient vector is a finite coefficient sequence with no zero entries and no equal consecutive entries. Its compression is obtained by first deleting zeros and then replacing each consecutive run of equal values by one value. For positive vectors, the zero-deletion step is vacuous.

For example, $(0,1,1,0,2,2,1)$ compresses to $(1,2,1)$, and $(1,0,1)$ compresses to $(1)$. Repeated values separated by a genuinely different value remain: $(1,2,1)$ is already compressed. Thus compression is a run operation, not replacement by the set of distinct entries.

The notion records coefficient patterns in Milliken-Taylor systems and their separation under finite colorings. It does not assert that replacing a vector by its compression leaves the exact generated set unchanged for a fixed sequence: $(1,1)$ requires two nonempty ordered index blocks, whereas $(1)$ permits a single block with one index. The canonical pattern and the precise block constraints are different data.^[ref]

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]