Perfect and nearly perfect separation dimension of complete and random graphs

The separation dimension of a hypergraph (Formula presented.) is the smallest natural number (Formula presented.) for which there is an embedding of (Formula presented.) into (Formula presented.), 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 (Formula presented.), the separation dimension of any (Formula presented.) -vertex (Formula presented.) -graph is (Formula presented.), the perfect separation dimension is much larger. In fact, no polynomial upper-bound for the perfect separation dimension of (Formula presented.) -uniform hypergraphs is known. In our first result we essentially resolve the case (Formula presented.), that is, graphs. We prove that the perfect separation dimension of (Formula presented.) is linear in (Formula presented.), up to a small polylogarithmic factor. In fact, we prove it is at least (Formula presented.) and at most (Formula presented.). Our second result proves that the perfect separation dimension of almost all graphs is also linear in (Formula presented.), 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 (Formula presented.) is w.h.p. (Formula presented.) for a wide range of values of (Formula presented.), including all constant (Formula presented.). Finally, we prove that significantly relaxing perfection to just requiring that any pair of disjoint edges of (Formula presented.) is separated the same number of times up to a difference of (Formula presented.) for some absolute constant (Formula presented.), still requires the dimension to be (Formula presented.). This is perhaps surprising as it is known that if we allow a difference of (Formula presented.), then the dimension reduces to (Formula presented.).