### Variant of an Expander graph: Probability that S random points cast a shadow/projection of size at most S/2 on each face of a cube.

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 uppe...Read more