\section{The Universal Flow}
\label{section:stone-cech-flow}

\begin{definition}[Greatest Ambit]
\label{definition:stone-cech-flow}
    Let $G$ be a discrete group, and
    \[
    \beta L: G \to \text{Aut}(\beta G) \quad \beta L_g(\fU) = g\fU
    \]

    then
    \begin{enumerate}
        \item $(\beta L: G \curvearrowright \beta G, 1_G)$ is an ambit.  
        \item[(U)] For any ambit $(G \curvearrowright X, x_0)$, there exists a unique morphism of pointed $G$-flows $\pi: (\beta G, 1_G) \to (X, x_0)$. Moreover, $\pi$ is surjective. 
    \end{enumerate}
    
    Any ambit $(Y, y_0)$ satisfying (U) is the \textbf{greatest $G$-ambit}. 
\end{definition}
\begin{proof}
    (1): Let
    \[
    L: G \to \text{Perm}(G) \quad L_g(h) = gh
    \]

    be the left translation action on itself. For each $g \in G$, the functoriality of the \hyperref[Stone-Čech compactification]{definition:stone-cech} implies that $L_g$ extends into a continuous map $\beta L_g: \beta G \to \beta G$. By uniqueness of continuous extensions, $\beta L_g(\fU) = g\fU$ for all $\fU \in \beta G$. In particular, $\text{Id}_{\beta G} = \beta L_{e}$. 
    
    Moreover, for any $g, h \in G$, $\beta L_{gh} = \beta L_{g} \circ \beta L_{h}$, so $g \mapsto L_g$ is a homomorphism. Finally, for any $g \in G$, 
    \[
    \beta L_{g^{-1}} \circ \beta L_{g} = \beta L_{1_G} = \beta L_{g} \circ \beta L_{g^{-1}}
    \]
    
    and $\beta L_g \in \text{Aut}(\beta G)$. 

    Since $\bracsn{L_g1_G|g \in G} = G$ is dense in $\beta G$, $(G \curvearrowright \beta G, 1_G)$ is an ambit. 

    (2): For each $g \in G$, let $\pi_0(g) = gx_0$, then $\pi_0 \in C(G; X)$, $\pi_0(1_G) = x_0$, and the following diagram commutes
    \[
    \xymatrix{
    G \ar@{->}[r]^{\pi_0} \ar@{->}[d]_{g} & X \ar@{->}[d]^{g} \\
    G \ar@{->}[r]_{\pi_0} & X
    }
    \]

    for all $g \in G$. By universal property of $\beta G$, $\pi_0$ admits a unique continuous extension $\pi \in C(\beta G; X)$. By uniqueness of continuous extensions, the following diagram commutes
    \[
    \xymatrix{
    \beta G \ar@{->}[r]^{\pi} \ar@{->}[d]_{g} & X \ar@{->}[d]^{g} \\
    \beta G \ar@{->}[r]_{\pi} & X
    }
    \]

    for all $g \in G$. Finally, since $\beta G$ is compact and $X$ is Hausdorff, $\pi(\beta G)$ is \hyperref[also compact]{proposition:compact-extensions} and \hyperref[hence closed]{proposition:compact-closed}. As $(X, x_0)$ is an ambit and $\pi(\beta G) \supset Gx_0$, $\pi(\beta G) \supset \ol{Gx_0} = X$. 
\end{proof}

\begin{definition}
\label{definition:stone-cech-semigroup}
    Let $G$ be a discrete group. For each $\fU, \fV \in \beta G$, let $\fU \cdot \fV = \lim_{g \to \fU}g\fV$, then
    \begin{enumerate}
        \item For each $A \subset G$, $A \in \fU \cdot \fV$ if and only if $\bracsn{g \in G|g^{-1}A \in \fV} \in \fU$. 
        \item $\beta G$ equipped with the composition law $(\fU, \fV) \mapsto \fU \cdot \fV$ is a semigroup with identity $1_G$. 
        \item (\textbf{Right Topological}) For each $\fV \in \beta G$, the mapping $\beta G \to \beta G$ defined by $\fU \mapsto \fU \cdot \fV$ is continuous. 
    \end{enumerate}
\end{definition}
\begin{proof}
    (3): By (U2) of the \hyperref[Stone-Čech compactification]{definition:stone-cech}. 
\end{proof}

\begin{definition}[Induced Action]
\label{definition:stone-cech-semigroup-induced-action}
    Let $G$ be a group, $(X, x_0)$ be a pointed $G$-flow, and $p_x: G \to X$ be the orbit map, then the mapping
    \[
    \pi: \beta G \to C(X; X) \quad \pi_\fU x  = \fU x = \lim_{g \to \fU}gx
    \]

    is a semigroup action of $\beta G$ on $X$, known as the \textbf{induced action} of $\beta G$ on $X$. 
\end{definition}



\begin{proposition}
\label{proposition:tds-return-set-stone-cech}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$. For each $y \in X$ and $\fU \in \beta G$, then $y = \fU x$ if and only if $N(x, U) \in \fU$.
\end{proposition}
\begin{proof}
    By definition of filter limits, $y = \fU x$ if and only if $p_x^{-1}(U) \in \fU$ for all $U \in \cn_X(y)$. Since $p_x^{-1}(U) = N(x, U)$, $y = \fU x$ if and only if $N(x, U) \in \fU$. 
\end{proof}

\begin{proposition}
\label{proposition:z-stone-cech-fixed-point}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then the following are equivalent:
    \begin{enumerate}
        \item For each $U \in \cn_X(x)$, $N(x, U)$ is infinite. 
        \item There exists $\fU \in \beta G \setminus G$ such that $\fU x = x$. 
    \end{enumerate}
\end{proposition}
% "obvious"


\begin{definition}[Left Ideal]
\label{definition:left-ideal-semigroup}
    Let $G$ be a semigroup and $\emptyset \ne I \subset G$, then $I$ is a \textbf{left ideal} if for any $g \in G$ and $h \in I$, $gh \in I$. The left ideal $I$ is \textbf{minimal} if there exists no left ideal $J \subsetneq I$. 
\end{definition}

\begin{proposition}
\label{proposition:stone-cech-ideal-subflow}
    Let $G$ be a group and $\emptyset \ne I \subset \beta G$. If $I$ is closed, then the following are equivalent:
    \begin{enumerate}
        \item $L$ is a $G$-subflow of $\beta G$. 
        \item $L$ is a closed left ideal of $\beta G$. 
    \end{enumerate}

    Moreover, for arbitrary $\emptyset \ne I \subset \beta G$, the following are equivalent
    \begin{enumerate}[label=(M\arabic*)]
        \item $L$ is a minimal $G$-subflow of $\beta G$.
        \item $L$ is a minimal left ideal of $\beta G$. 
    \end{enumerate}
    
    In particular, minimal left ideals of $\beta G$ are closed. 
\end{proposition}

\begin{corollary}
\label{corollary:stone-cech-minimal-ideal-existence}
    Let $G$ be a group, then $\beta G$ admits a minimal ideal. 
\end{corollary}
\begin{proof}
    By \autoref{theorem:tds-minimal-existence} and \autoref{proposition:stone-cech-ideal-subflow}. 
\end{proof}

\begin{definition}[Universal Minimal Flow]
\label{definition:universal-minimal-flow}
    Let $G$ be a group, then there exists a flow $L$ such that:
    \begin{enumerate}
        \item $G \curvearrowright L$ is a minimal $G$-flow. 
        \item[(U)] For any minimal $G$-flow $X$, there exists a continuous map $\Phi \in C(M; X)$ such that $\Phi(m_0) = x_0$ and the following diagram commutes
        \[
        \xymatrix{
        L \ar@{->}[r]^{\Phi} \ar@{->}[d]_{g} & X \ar@{->}[d]^{g} \\
        L \ar@{->}[r]_{\Phi} & X
        }
        \]

        for all $g \in G$.
    \end{enumerate}

    The flow $(M, \alpha)$ is the \textbf{universal minimal $G$-flow}.
\end{definition}
\begin{proof}
    (1): Using \autoref{corollary:stone-cech-minimal-ideal-existence}, let $L \subset \beta G$ be a minimal left ideal, then $G \curvearrowright L$ is a minimal $G$-flow. 

    (U): Let $G \curvearrowright X$ be a minimal $G$-flow, $x_0 \in X$, and $m_0 \in M$. By (U) of the \hyperref[universal flow]{definition:stone-cech-flow}, there exists a unique continuous map $\phi \in C(\beta G; X)$ such that $\phi(1_G) = x_0$ and the following diagram commutes
    \[
    \xymatrix{
    \beta G \ar@{->}[r]^{\Phi} \ar@{->}[d]_{g} & X \ar@{->}[d]^{g} \\
    \beta G \ar@{->}[r]_{\Phi} & X
    }
    \]

    for all $g \in G$. 

    Let $\Phi = \phi|_{L} \in C(L; X)$, then as $L$ is $G$-invariant, $\Phi(L) \supset G\phi(x_0)$. By \autoref{proposition:compact-extensions}, $\Phi(L) = \ol{G\phi(x_0)} = X$.  
\end{proof}


\begin{definition}[Minimal Idempotent]
\label{definition:tds-minimal-idempotent}
    Let $S$ be a compact Hausdorff right topological semigroup and $e \in S$ be an idempotent, then $e$ is \textbf{minimal} if there exists a minimal left ideal $I \subset G$ such that $e \in I$. 
\end{definition}

\begin{theorem}[Ellis]
\label{theorem:ellis-semigroup}
    Let $S$ be a non-empty compact Hausdorff right topological semigroup and $I \subset S$ be a closed left ideal, then there exists a minimal left ideal $I \subset S$ and a minimal idempotent $e \in I$. 
\end{theorem}
\begin{proof}
    By Zorn's Lemma, $I$ admits a minimal non-empty closed right topological subsemigroup $K \subset S$. For each $a \in K$, $Ka$ is closed by continuity of right multiplication and \autoref{proposition:compact-extensions}. By minimality of $K$, $K = Ka$, so there exists $e \in K$ such that $a = ea$. Let $E = \bracsn{b \in K|ba = a}$, then $E$ is also a closed and right topological subsemigroup. By minimality of $K$, $E = K$, and in particular $a \in E$. 
\end{proof}

\begin{corollary}
\label{corollary:stone-cech-minimal-idempotent}
    Let $G$ be an infinite group, then $\beta G$ admits a minimal idempotent in $\beta G \setminus G$. 
\end{corollary}
\begin{proof}
    Let $L \subset (\beta G \setminus G)$ be a minimal left ideal, then $L$ is a non-empty compact Hausdorff right topological semigroup. By \hyperref[Ellis' Theorem]{theorem:ellis-semigroup}, there exists $e \in L$ with $e^2 = e$. 
\end{proof}

\begin{proposition}
\label{proposition:stone-cech-flow-minimal-idempotent}
    Let $G$ be an infinite group, $G \curvearrowright X$ be a minimal $G$-flow, then there exists a minimal idempotent $\fU \in \beta G \setminus G$ such that $\fU x = x$. 
\end{proposition}
\begin{proof}
    Let $L \subset \beta G$ be a minimal left ideal, then the mapping $\fU \mapsto \fU x$ is a continuous surjection. As $x \in X = \bracsn{\fU x|x \in L}$, there exists $\fU \in L$ with $\fU x = x$. Let $F = \bracsn{\fU \in L|\fU x = x}$, then $F$ is a non-empty closed right topological subsemigroup of $L$. By \hyperref[Ellis' Theorem]{theorem:ellis-semigroup}, $F$ contains an idempotent. 
\end{proof}


\begin{theorem}[Infinite Ramsey Theorem for Pairs]
\label{theorem:ramsey-pairs}
    Let
    \[
    [\natp]^2 = \bracsn{\bracsn{m, n} \in \nat^2|m, n \in \natp, m \ne n}
    \]
    
    be the set of unordered pairs of distinct elements of $\natp$ and $c: [\natp]^2 \to \bracsn{1, \cdots, r}$, then there exists $A \subset \natp$ infinite such that for any $m, n, m', n' \in A$, $c(\bracsn{m, n}) = c(\bracsn{m', n'})$. 
\end{theorem}
\begin{proof}
    Let $\fU \subset 2^{\natp}$ be a free ultrafilter. For each $n \in \natp$ and $1 \le i \le r$, let $A(n, i) = \bracsn{m > n| c(\bracsn{m,n}) = i}$, then $\natp = \bracsn{1, \cdots, n} \bigsqcup_{i = 1}^r A(n, i)$. As $\fU$ is a free ultrafilter, there exists a unique $1 \le i(n) \le r$ such that $A(n, i(n)) \in \fU$. 

    For each $1 \le i \le r$, let $B_i = \bracsn{n \in \natp|i(n) = i}$, then $\natp = \bigsqcup_{i = 1}^r B_i$. As $\fU$ is an ultrafilter, there exists a unique $1 \le i \le r$ with $B_i \in \fU$. 
    
    Let $n_1 \in B_i$, so $i(n_1) = i$. Assume inductively that $\bracsn{n_j}_1^N \subset \natp$ has been defined such that:
    \begin{enumerate}[label=(\roman*)]
        \item For each $1 \le j \le N - 1$, $n_j < n_{j+1}$. 
        \item For each $1 \le j \le N$, $i(n_j) = i$. 
        \item For each $1 \le j < k \le N$, $c(\bracsn{n_j, n_k}) = i$. 
    \end{enumerate}

    Let $n_{N+1} \in B_i \cap \bigcap_{j = 1}^N A(n_j, i)$, then:
    \begin{enumerate}
        \item As $n_{N+1} \in A(n_N, i)$, $n_{N+1} > n_N$. 
        \item Since $n_{N+1} \in B_i$, $i(n_{N+1}) = i$. 
        \item For each $1 \le j \le N$, $n_{N+1} \in A(n_j, i) = A(n_j, i(n))$, so $c(\bracsn{n_j, n_{N+1}}) = i$. 
    \end{enumerate}

    By induction, there exists $\seq{n_j} \subset \natp$ such that $c(\bracsn{n_j, n_k}) = i$ for all $j, k \in \natp$ with $j \ne k$. 
\end{proof}


\begin{lemma}[Idempotent Refinement]
\label{lemma:idempotent-refinement}
    Let $S$ be a semigroup and $f \in \beta S$ be idempotent, then there exists a minimal idempotent $e \in \beta S$ such that $ef = fe = e$. 
\end{lemma}
\begin{proof}
    By \autoref{proposition:compact-extensions}, $(\beta S)f$ is a closed left ideal in $S$. By \hyperref[Ellis' Theorem]{theorem:ellis-semigroup}, $(\beta S)f$ contains a minimal left ideal $L$, and a minimal idempotent $r \in L$. As $L \subset (\beta S)f$, there exists $t \in \beta S$ such that $r = tf$. 

    Let $e = fr \in L$, then $e^2 = frfr = fr^2 = fr = e$. Moreover, 
    \[
    ef = frf = ftf^2 = ftf = fr = e = fr = ffr = fe
    \]
\end{proof}