Separation dimension and sparsity

The separation dimension 𝜋π(G) of a hypergraph G is the smallest natural number k for which the vertices of G can be embedded in R^k so that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the cardinality of a smallest family F of total orders of V(G), such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. Separation dimension is a monotone parameter; adding more edges cannot reduce the separation dimension of a hypergraph. In this article, we discuss the influence of separation dimension and edge-density of a graph on one another. On one hand, we show that the maximum separation dimension of a k-degenerate graph on n vertices is O(k lg lg n) and that there exists a family of 2-degenerate graphs with separation dimension Ω(lg lg n). On the other hand, we show that graphs with bounded separation dimension cannot be very dense. Quantitatively, we prove that n-vertex graphs with separation dimension s have at most 3(4 lg n)^s−2 edges. We do not believe that this bound is optimal and give a question and a remark on the optimal bound.