The answer is $k=23$. Example: Take $24$ clubs($c_1,c_2,\dots,c_{24}$) and students $S_1,S_2,\dots,S_{1200}$ such that $S_i$ belongs to all the clubs except $c_j$ (with $i\equiv j$ mod 24). For any union of $23$ students at least one club can not be withdrawn. For $k\leq 22$ take two students $S_i$ and $S_j$ who share at most $21$ clubs. Define $c_1,c_2,\dots,c_{21}$ as the common clubs, notice that the "common club" must be one of them. Choose the following list of $21$ students: $\blacksquare$ The student who does not belong to $c_1$. $\blacksquare$ The student who does not belong to $c_2$. $\dots$ $\blacksquare$ The student who does not belong to $c_{21}$. It's easy to see that these $23$ students does not share a "common club".