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Density Conditions For Triangles In Multipartite Graphs

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We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erdős, we prove that, if the minimum degree of G is at least 9|V (G)|/10, the largest triangle-free subgraphs are precisely the largest bipartite subgraphs in G. We investigate in particular the case where G is a complete multipartite graph. We prove that a finite tripartite graph with all edge densities greater than the golden ratio has a triangle and that this bound is best possible. Also we show that an infinite-partite graph with finite parts has a triangle, provided that the edge density between any two parts is greater than 1/2.

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Correspondence to Adrian Bondy.

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Bondy, A., Shen, J., Thomassé, S. et al. Density Conditions For Triangles In Multipartite Graphs. Combinatorica 26, 121–131 (2006).

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