%! TEX root = Combi.tex
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% 10/10/2024 09AM

% Plan:
% Chapter 1: Set systems
% Chapter 2: Isoperimetric inequalities
% Chapter 3: Intersecting families

% 3 examples sheets / 3 examples classes

% Books:
% 1. Combinatorics, Bollob\'as, C.U.P (1986).
%    Covers chapters 1 and 2. Very clear, easy to
%    read.
% 2. Combinatorics of finite sets, Anderson, O.U.P
%    (1987). Chapter 1.

% Course = what is on the board

\newpage
\section{Set Systems}

\begin{fcdefn}[Set system]
  \glsnoundefn{ssystem}{set system}{set systems}
  Let $X$ be a set. A \emph{set system} on $X$ (or
  a \emph{family of subsets of $X$}) is a family
  $\mathcal{A} \subset \powset X$.
\end{fcdefn}

\begin{notation*}
  \glsnoundefn{rset}{$r$-set}{$r$-sets}
  We will use the notation
  \[ X^{(r)} = \{A \subset X : |A| = r\} .\]
  We call an element of $X\rsubs$ an $r$-set.
  We will usually be using $X = [n] = \{1, \ldots,
  n\}$, so $|X\rsubs| = {n \choose r}$.
\end{notation*}

\begin{example*}
  \[ [4]\rsubs[2] = \{12, 13, 14, 23, 24, 34\} .\]
\end{example*}

\begin{fcdefn}[Discrete cube]
  \glsnoundefn{dcube}{discrete cube}{discrete
  cubes}
  \glssymboldefn{dcubeQn} %{$Q_n$}{$Q_n$}
  Make $\powset X$ into a graph by joining $A$ and
  $B$ if $|A \symmdiff B| = 1$, i.e. if $A = B
  \cup \{i\}$ for some $i$, or vice versa. We call
  this ths \emph{discrete cube} $Q_n$ (if $X =
  [n]$).
\end{fcdefn}

\begin{example*}
  $\Q_3$:
  \begin{center}
    \includegraphics[width=0.2\linewidth]{images/7574d50a51ab4407.png}
  \end{center}
  In general:
  \begin{center}
    \includegraphics[width=0.7\linewidth]{images/ace31118117a4ae4.png}
  \end{center}
\end{example*}

Alternatively, can view $\Q_n$ as an
$n$-dimensional unit cube $\{0, 1\}^n$, by
identifying e.g. $\{1, 3\}$ with $1010000\ldots 0$
(i.e. identify $A$ with $\indicator{A}$, the
characteristic function of $A$).

\begin{example*}
  \phantom{}
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/c404887130cf4215.png}
  \end{center}
\end{example*}

\begin{fcdefn}[Chain]
  \glsnoundefn{chain}{chain}{chains}
  Say $\mathcal{A} \subset \powset X$ is a
  \emph{chain} if $\forall A, B \in \mathcal{A}$,
  either $A \subset B$ or $B \subset A$.
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.2\linewidth]{images/f4aebc423eac42a2.png}
\end{center}

\begin{example*}
  For example,
  \[ \mathcal{A} = \{23, 12357, 123567\} \]
  is a \gls{chain}.
\end{example*}

\begin{fcdefn}[Antichain]
  \glsnoundefn{antichain}{antichain}{antichains}
  Say $\mathcal{A}$ is an \emph{antichain} if
  $\forall A, B \in \mathcal{A}$, $A \neq B$, we
  have $A \not\subset B$.
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.2\linewidth]{images/7b8832f22baf45ff.png}
\end{center}

How large can a \gls{chain} be?
Can achieve $|\mathcal{A}| = n + 1$, for example
using
\[ \mathcal{A} = \{\emptyset, 1, 12, 123, \ldots,
[n]\} .\]
Cannot beat this: for each $0 \le r \le n$,
$\mathcal{A}$ contains $\le 1$ \gls{rset}.

How large can an \gls{antichain} be?
Can achieve $|\mathcal{A}| = n$, for example
$\mathcal{A} = \{1, 2, \ldots, n\}$. More
generally, can take $\mathcal{A} = X\rsubs$, for
any $r$ -- best out of these is
$X\rsubs[\left\lfloor \frac{n}{2} \right\rfloor]$.

Can we beat this?