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On the existence of triangulated spheres in 3-graphs, and related problems. (English) Zbl 0269.05111
The problem described in the title represents an analogue of the well known property of graphs that any graph on $$n$$ vertices and having at least $$n$$ edges contains a polygon. That result could be restated, in topological terms, as saying that any simplicial 1-complex with at least as many 1-simplexes as 0-simplexes must contain a triangulation of the 1-sphere. In Theorem 3 we shall determine asymptotically the maximum number of 2-simplexes a simplicial 2-complex may contain without containing a subcomplex which is a triangulation of the 2-sphere. More precisely, we shall prove that there exist constants $$c_1$$ and $$c_2$$ such that every 3-graph on $$n$$ vertices having $$c_2n^{3/2}$$ edges or more contains a double pyramid; but that there exists a 3-graph on $$n$$ vertices having $$c_1n^{3/2}$$ edges containing no triangulation of the sphere. Also, we discuss several related results.
Show Scanned Page ##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 57M20 Two-dimensional complexes (manifolds) (MSC2010) 05C35 Extremal problems in graph theory
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