The other day, I met with professor Halfbrain and professor Erasmus in the coffee house. Professor Erasmus told us that he had been working on a schedule for a tennis club with $30$ senior and $30$ junior members.
- Every senior member should play against one other senior member once, and against $15$ of the junior members once.
- Every junior member should play against one other junior member once, and against $15$ of the senior members once.
- No further matches should be scheduled.
Professor Halfbrain listened to this attentively, and then claimed that in each such schedule there will be two senior members $S_1$ and $S_2$ and two junior members $J_1$ and $J_2$ such that:
- The two senior members $S_1$ and $S_2$ play against each other.
- The two junior members $J_1$ and $J_2$ play against each other.
- Each of $S_1$ and $S_2$ play against at least one of $J_1$ and $J_2$.
- Each of $J_1$ and $J_2$ play against at least one of $S_1$ and $S_2$.
Question: Is professor Halfbrain's claim indeed true, or has the professor once again made one of his mathematical blunders?