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 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. 
 A: You're essentially asking if you drop $k$ objects each into one of $k$ bins, what is the probability that only $k/2$ bins are occupied. (You can forget the extra dimension and only think about the dimensions you're looking at). 
If you let $E_i$ be the event that the $i$th bin is empty, this has probability $((k-1)/k)^k\approx 1/e$. The events $E_i$ are approximately independent. Letting $N=\sum_{i=1}^k \mathbf 1_{E_i}$, $N$ has expectation $k/e$. The probability that there are $k/2$ unoccupied bins will decay as $e^{-\alpha k}$ using large deviation estimates. 
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If you want to pick a subset of size $\alpha k$ of the set of size $k$, it's obvious how to do it: take the boxes with the most elements first. You expect the distribution of elements in each box to be approximately Poisson with parameter 1. This means that approximately $(1-2/e)k$ of the boxes have two or more elements. As long as $\alpha$ is less than $1-2/e$, there's a high probability that you can find a subset $S$ of size $\alpha k$ whose shadow has size at most $|S|/2$.
