Middle European Mathematical Olympiad 2023 Problem T-4

Let $c \geq 4$ be an even integer. In some football league, each team has a home uniform and an away uniform. Every home uniform is coloured in two different colours, and every away uniform is coloured in one colour. A team's away uniform cannot be coloured in one of the colours from the home uniform. There are at most $c$ distinct colours on all of the uniforms. If two teams have the same two colours on their home uniforms, then they have different colours on their away uniforms.

We say a pair of uniforms is clashing if some colour appears on both of them. Suppose that for every team $X$ in the league, there is no team $Y$ in the league such that the home uniform of $X$ is clashing with both uniforms of $Y$. Determine the maximum possible number of teams in the league.