Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly random locations, i.e., for each point choose its co-ordinates uniformly at random $(r_0,r_1,r_2)$ where $ 0 \leq r_i < \sqrt{k}$.

For an arbitrarily small number $\alpha >0$. Let $E$ be an event that there exists a subset of points of size $S = \alpha k$ such that the projection of these S points on every face of the cube, is of size at most $S/2$.

What is an upper bound on the probability of event $E$, i.e., P(E)?

Caution: If you start with a fixed set of size $S$, you would need to consider all possible subsets of $k$ points of size $S$. The event $E$ is defined as **there exists a subset of size $S$ in $k$ points which are randomly placed.**