VN-TST 1991 Problem 6

Let a set $X$ be given which consists of $2 \cdot n$ distinct real numbers ( $n \geq 3$ ). Consider a set $K$ consisting of some pairs $(x, y)$ of distinct numbers $x, y \in X$ , satisfying the two conditions:

I. If $(x, y) \in K$ then $(y, x) \not \in K$ . II. Every number $x \in X$ belongs to at most 19 pairs of $K$ .

Show that we can divide the set $X$ into 5 non-empty disjoint sets $X_1, X_2, X_3, X_4, X_5$ in such a way that for each $i = 1, 2, 3, 4, 5$ the number of pairs $(x, y) \in K$ where $x, y$ both belong to $X_i$ is not greater than $3 \cdot n$ .