1987 IMO 第 2 题

(USA 3) At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_{1}, C_{2}, \ldots, C_{k}$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2 n$.

(美国 3)在 $n$ 已婚夫妇参加的聚会上，每个人都与聚会上除他或她的配偶之外的其他人交谈。对话涉及一组人或派系 $C_{1}, C_{2}, \ldots, C_{k}$ ，具有以下属性：没有一对是同一派系的成员，但对于每对其他人来说，两个成员都属于一个派系。证明如果$n \geq 4$，则$k \geq 2 n$。