Example 1: 10 college students organize sports teams according to the following conditions:
(1) Each person can sign up for several sports teams at the same time;
(2) No team can be completely contained within another team or overlap entirely with another team (but partial overlap is allowed).
Under these two conditions, what is the maximum number of teams that can be organized? How many people are in each team?
Let $M \triangleq\{$ the set of sports teams that satisfy conditions (1), (2), and contain the maximum number of teams $\}$.
$$
M_{i} \in M,\left|M_{i}\right|=i \text {, and } r=\min \left\{i \mid M_{i} \neq \varnothing\right\}, s=\max \left\{i \mid M_{i} \neq \varnothing\right\} \text {. }
$$
(1) If $s>5$, let $N \triangleq\{M$, all possible sports teams obtained by removing one athlete\}, then $|N|=s-1$, so for any $A \in M_{s}$, there exists $B_{j} \in N, 1 \leqslant j \leqslant s$, such that $B_{j} \subseteq A, 1 \leqslant j \leqslant s$ ($B_{j}$ is obtained by removing one person from $A$); and for each $B \in N$, there are no more than $11-s$ $A_{j}, 1 \leqslant j \leqslant 11-s$, such that $B \subseteq A_{j}$ (adding at most $10-(s-1)=11-s$ people from the teams not in $N$ might result in a team not in $M_{s}$), thus $(11-s)|N| \geqslant s\left|M_{s}\right|$, hence
$$
\begin{array}{l}
|N| \geqslant \frac{s}{11-s}\left|M_{s}\right| \geqslant \frac{6}{5}\left|M_{s}\right|>\left|M_{s}\right|, \\
\begin{aligned}
\sum_{j=r}^{s-1}\left|M_{j}\right|+|N|> & \sum_{j=r}^{s-1}\left|M_{j}\right|+\left|M_{s}\right| \\
& =\sum_{j=r}^{s}\left|M_{j}\right|=|M| .
\end{aligned}
\end{array}
$$

Next, we prove that $M_{j}(r \leqslant j \leqslant s-1), N$ all satisfy conditions (1), (2). Condition (1) is obviously satisfied; for condition (2), if there exists $X \in M_{i}$ and $X \in N$, by the definition of $N$, there exists a $Y \in M_{s}$ such that $X \subseteq Y$, which contradicts the definition of $M$. Also, note that for any $P \in N, Q \in M_{i}$, we have $|P| \geqslant|Q|$, so it cannot be that $P \subseteq Q$, which contradicts the assumption of the maximality of $M$, hence $s \leqslant 5$.
(2) Similarly, we can prove $r \geqslant 5$, thus $r=s=5$, meaning all teams consist of 5 people. The number of teams consisting of 5 people is $\mathrm{C}_{10}^{5}$, which clearly satisfies conditions (1), (2), so the maximum number of teams is $\mathrm{C}_{10}^{5}=252$, each containing 5 people.