Problem: Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers.
In a group of $t_n+1$ people, some games of chess are played.
Two people can play each other at most once.
Prove that it is possible for the following two conditions to hold at the same time:
\begin{enumerate}[(i)]
  \ii The number of games played by each person is one of $t_1,t_2,\dots,t_n$.
  \ii For every $i$ with $1\leq i\leq n$,
  there is someone who has played exactly $t_i$ games of chess.
\end{enumerate}

Solution: Phrased in graph theory,
the problem asks to produce a simple graph $G$ on $t_n+1$ vertices
such that all degrees are in the set $\{t_1, \dots, t_n\}$
and each degree appears at least once.

We proceed by induction on $n$.
If $n = 1$, take a clique on $t_1+1$ vertices.
For $n = 2$, take a clique on $t_1$ vertices and
an empty graph on $t_2 + 1 - t_1$ vertices, and join them all together.
For the inductive step:
\begin{itemize}
  \ii Take an example for the $(n-2)$ tuple $(t_2-t_1, \dots, t_{n-1}-t_1)$,
  which has $t_{n-1} - t_1 + 1$ vertices.
  \ii Then add in $t_n - t_{n-1}$ isolated vertices.
  \ii Finally add in $t_1$ universal vertices.
\end{itemize}

\begin{remark*}
The universal vertices are ``forced'',
so the only parameter is the number of universal vertices to add in.
\end{remark*}