There are 1991 participants in a sports event. Each of them knows at least $n$ other participants (the relationship of knowing each other is reciprocal). What is the minimum value of $n$ for which there must necessarily exist a group of 6 participants who know each other pairwise?
We construct a graph where the vertices are the participants, connected by an edge when they know each other. Then every vertex has a degree of at least $n$, and we seek the minimal value of $n$ for which this graph necessarily contains $K_{6}$. According to Zarankiewicz's lemma, a 6-free graph with 1991 vertices contains at least one vertex with a degree less than or equal to $\tilde{\mathrm{A}}\left\lfloor\frac{4 \times 1991}{5}\right\rfloor=1592$. Therefore, the bound $n \geq 1593$ is sufficient. On the other hand, this bound is optimal if we consider the 6-free Turán graph $T(1991,5)$. This graph is the complete 5-partite graph with 399 edges in one of the parts and 398 in the others. The minimal degree of a vertex in this graph is consequently $1991-399=1592$.