$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other.
Show that $n \leq \frac{p^{2}}4$.
1. **Model the problem using graph theory:**
   - Represent the chess tournament as a graph \( G \) where each vertex represents a player and each edge represents a game played between two players.
   - Given that in every subset of three players, at least two have not played against each other, this implies that \( G \) is triangle-free (i.e., it contains no \( K_3 \) subgraph).

2. **Apply Mantel's Theorem:**
   - Mantel's Theorem states that the maximum number of edges in a triangle-free graph with \( p \) vertices is \( \left\lfloor \frac{p^2}{4} \right\rfloor \).
   - Since \( G \) is triangle-free, the number of edges \( n \) in \( G \) must satisfy \( n \leq \left\lfloor \frac{p^2}{4} \right\rfloor \).

3. **Conclude the proof:**
   - Since \( p \) is an integer and \( p > 2 \), the floor function \( \left\lfloor \frac{p^2}{4} \right\rfloor \) simplifies to \( \frac{p^2}{4} \) when \( p \) is even, and \( \frac{p^2 - 1}{4} \) when \( p \) is odd.
   - Therefore, in both cases, \( n \leq \frac{p^2}{4} \).

\[
\boxed{n \leq \frac{p^2}{4}}
\]