Pii: S0012-365x(98)00309-4

نویسندگان

Geir Agnarsson

William T. Trotter

چکیده

With a finite graph G = (V,E), we associate a partially ordered set P = (X, P) with X = V U 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, Tur~n's theorem and the Erd6s/Stone theorem to show that l i m p ~ M (p ,4 ) /p2= 3/8. We then derive some ring theoretic consequences of this in terms of minimal first syzygies and Betti numbers for monomial ideals. (~) 1999 Elsevier Science B.V. All rights reserved

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

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