KöMaL 2025 C1844

Ági uses red and Laci uses blue to color the squares in an $n\times n$ white board. Let $(i,j)$ denote the square in the $i^{\text{th}}$ row and $j^{\text{th}}$ coloumn. In the first round Ági colors the squares in the main diagonal (from the top left to the bottom right) red. Now they take turns: if Laci colors square $(i,j)$ blue, then Ági colors $(j,i)$ red. They color every square exactly once. The $k^{\text{th}}$ row is special if for every blue $(k,j)$ there exists $l$ such that both $(k,l)$ and $(l,j)$ is red. Prove that after finishing the coloring Ági is guaranteed to find a special row.