Question 6
Let $X$ be a set of $2n$ distinct real numbers, where $n\ge3$. Let $K$ be a set of ordered pairs $(x,y)$ with $x,y\in X$ and $x\ne y$, satisfying: (1) if $(x,y)\in K$, then $(y,x)\notin K$; (2) each element $x\in X$ occurs in at most 19 pairs of $K$. Prove that $X$ can be partitioned into five nonempty pairwise disjoint subsets $X_1,\ldots,X_5$ such that, for each $i=1,\ldots,5$, at most $3n$ pairs $(x,y)\in K$ have both $x,y\in X_i$.