SCHWEITZER 2008 Problem 2

Let $t\ge 3$ be an integer, and for $1\le i <j\le t$ let $A_{ij}=A_{ji}$ be an arbitrary subset of an $n$ -element set $X$ . Prove that there exist $1\le i < j\le t$ for which $$\left| \left( X\,\backslash A_{ij}\right) \cup \bigcup_{k\neq i,j}\left( A_{ik}\cap A_{jk}\right) \right| \ge \frac{t-2}{2t-2}n$$ (translated by Miklós Maróti)