Does every Latin square contain a diagonal in which no symbol appears thrice?

Question

A diagonal of a Latin square is a selection of n entries in which no two entries occur in the same row or column. For example: the entries marked with an asterisk below form a diagonal.

1  2* 3  4
2  3  4  1*
3  4  1* 2
4* 1  2  3

Theorem: Every Latin square contains a diagonal in which no symbol appears thrice (or more).

The asterisked diagonal in the above example is a diagonal in which no symbol appears thrice.

Problem: Prove the above theorem.

This is quite a fun problem to solve, but there is a trap.

Answer 1

You can do better- Cameron and Wanless showed that every latin square possesses a diagonal in which no symbol appears more than twice.

We also show that every Latin square contains a set of entries which meets each row and column exactly once while using no symbol more than twice.

For the paper, see Covering radius for sets of permutations