Perfect and nearly perfect separation dimension of complete and random graphs

The separation dimension of a hypergraph $G$ is the smallest natural number $d$ for which there is an embedding of $G$ into $\mathbb{R}^d$, such that any pair of disjoint edges is separated by some hyperplane normal to one of the axes. The perfect separation dimension further requires that any pair of disjoint edges is separated by the same amount of such (pairwise nonparallel) hyperplanes. While it is known that for any fixed $r \ge 2$, the separation dimension of any $n$-vertex $r$-graph is $O(\log n)$, the perfect separation dimension is much larger. In fact, no polynomial upper-bound for the perfect separation dimension of $r$-uniform hypergraphs is known. In our first result we essentially resolve the case $r=2$, i.e. graphs. We prove that the perfect separation dimension of $K_n$ is linear in $n$, up to a small polylogarithmic factor. In fact, we prove it is at least $n/2-1$ and at most $n(\log n)^{1+o(1)}$. Our second result proves that the perfect separation dimension of almost all graphs is also linear in $n$, up to a logarithmic factor. This follows as a special case of a more general result showing that the perfect separation dimension of the random graph $G(n,p)$ is w.h.p. $\Omega(n p /\log n)$ for a wide range of values of $p$, including all constant $p$. Finally, we prove that significantly relaxing perfection to just requiring that any pair of disjoint edges of $K_n$ is separated the same number of times up to a difference of $c \log n$ for some absolute constant $c$, still requires the dimension to be $\Omega(n)$. This is perhaps surprising as it is known that if we allow a difference of $7\log_2 n$, then the dimension reduces to $O(\log n)$.