Limits of dense graph sequences
Journal of Combinatorial Theory Series B
Random Structures & Algorithms
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A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌋n2/4⌋; + 1 edges. Rademacher proved that G contains at least ⌋n/2⌋; triangles, and several authors proved that G contains a book of size at least n/6. We prove the following “linear combination” of these two results. Suppose that and the maximum size of a book in G is less than αn/2. Then G contains at least triangles as n→Ȟ. This is asymptotically sharp. On the other hand, for every , there exists β0 such that G contains at least βn3 triangles. It remains an open problem to determine the largest possible β in terms of α. Our short proof uses the triangle removal lemma, although there is another approach which avoids this. © 2011 Wiley Periodicals, Inc. J Graph Theory (Contract grant sponsor: NSF; Contract grant number: DMS-0653946. © 2012 Wiley Periodicals, Inc.)