= Dynamical proof of Hindman's theorem
Extend a <finite coloring> of the <positive integers> to a point $x$ of a two-sided <full shift>. A <minimal subsystem> of its forward <orbit closure>, together with the proximal-minimal existence theorem, supplies a <minimal point> $y$ <proximal> to $x$. Put $q=y(0)$. If all sums in $\{0\}\cup\operatorname{FS}(a_1,\ldots,a_r)$, the augmented <finite-sums set>, have color $q$ in $y$, their coordinate constraints define a <cylinder set> containing $y$. The <joint return lemma for a proximal minimal pair> chooses a new positive term so that all new sums have color $q$ in both $x$ and $y$. <Mathematical induction> gives an infinite <monochromatic> <finite-sums set>. The new term can exceed the sum of all previous terms, giving unique representations. The proximal-minimal existence theorem is a substantive input to this proof.
Back to article page
