The maximum number of edges in a graph of bounded dimension, with applications to ring theory

AbstractWith a finite graph G = (V, E), we associate a partially ordered set P = (X, P) with X = V ∪ E and x < e in P if and only if x is an endpoint of e in G. This poset is called the incidence poset of G. In this paper, we consider the function M(p, d) defined for p, d ⩾ 2 as the maximum number of edges a graph G can have when it has p vertices and the dimension of its incidence poset is at most d. It is easy to see that M(p, 2) = p − 1 as only the subgraphs of paths have incidence posets with dimension at most 2. Also, a well known theorem of Schnyder asserts that a graph is planar if and only if its incidence poset has dimension at most 3. So M(p, 3) = 3 p − 6 for all p ⩾ 3. In this paper, we use the product ramsey theorem, Turán's theorem and the Erdo&#x030B;s/Stone theorem to show that limp→∞ M(p, 4)/p2 = 38. We then derive some ring theoretic consequences of this in terms of minimal first syzygies and Betti numbers for monomial ideals