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

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Abstract

With a finite graph G = (V, E), we associate a partially ordered set P = (X, P) with X = VE 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.

Keywords

Graph
Partially ordered set
Dimension
Regularity lemma
Ramsey theory
Extremal graph theory

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Research supported in part by the Office of Naval Research and the Deutsche Forschungsgemeinschaft.