Hindman (Furstenberg) [latex] \begin{theorem}[Hindman] \label{thm:hindman} \label{thm:hindmanfurstenberg} If $\N$ is partitioned into finitely many sets, then there is is an infinite subset $H \subseteq \N$ such that all finite sums of distinct elements of $H$ belong to the same set of the partition. \end{theorem} [/latex] Use: [latex] \begin{theorem} \label{thm:unifrprox} Let $X$ be a compact Hausdorff space and $T\colon X \to X$ continuous. Consider $(X,T)$.%TODO different notations Then for every $x \in X$ there is a uniformly recurrent $y \in X$ such that $y $ is proximal to $x$. \end{theorem} [/latex] [latex] \begin{refproof}{thm:hindmanfurstenberg}[Furstenberg] \begin{itemize} \item View partition as $f\colon \N \to k$. Consider $X \coloneqq k^{\N}$ (product topology, compact and Hausdorff). Let $x \in X$ be the given partition. \item $T\colon X \to X$ shift: $T(y)(n) \coloneqq y(n+1)$. \item Let $y$ proximal to $x$, uniformly recurrent. \begin{itemize} \item proximal $\leadsto$ $\forall N$.~$T^n(x)\defon_N = T^n(y)\defon_N$ for infinitely many $n$. \item uniform recurrence $\leadsto$ \[ \forall n .~\exists N.~\forall r.y\defon{\{r,\ldots,r+N-1\}} \text{ contains } $y\defon{\{0,\ldots,n\}}$ \text{ as a subsequence.} \] (consider neighbourhood $G_n = \{z \in X : z\defon{n} = y\defon{n} \}$). \end{itemize} \item Consider $c \coloneqq y(0)$. This color works: \begin{itemize} \item $G_0 \coloneqq y\defon{\{0\}}$, take $N_0$ such that $y\defon{\{r, \ldots, r + N_0 - 1\}} $ contains $y(0)$ for all $r$ (unif.~recurrence). $y\defon{\{r,\ldots,r+N_0 - 1\} } = x\defon{\{r,\ldots,r+N_0 -1\} }$ for infinitely many $r$ (proximality). Fix $h_0 \in \N$ such that $x(h_0) = y(0)$. \item $G_1 \coloneqq y\defon{\{0,\ldots,h_0\} }$, take $N_1$ such that $y\defon{\{r,\ldots,r +N_1-1\}}$ contains $y\defon{\{0,\ldots,h_0\} }$ for all $r$ (unif.~recurrence). So among ever $N_1$ terms, there are two of distance $h_0$ where $y$ has value $c$. So $\exists h_1 > h_0$ such that $x(h_1) = x(h_1 + h_0) = c$ (proximality). \item Repeat: Choose $h_i$ such that for all sums $s$ of subsets of $\{h_0,\ldots, h_{i-1}\}$, $x(s+h_i) = y(s+h_i) = c$: Find $N_i$ such that every $N_i$ consecutive terms of $y$ contain a segment that coincides with the initial segment of $y$ up to the largest $s$, then find a segment of length $N_i$ beyond $h_{i-1}$ where $x$ and $y$ coincide. \end{itemize} \end{itemize} \end{refproof} [/latex]