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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 269.05111

**Autor: ** Sos, V.T.; Erdös, Paul; Brown, William G.

**Title: ** On the existence of triangulated spheres in 3-graphs, and related problems. (In English)

**Source: ** Period. Math. Hung. 3, 221-228 (1973).

**Review: ** 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_{2}n^{3/2} edges or more contains a double pyramid; but that there exists a 3-graph on n vertices having c_{1}n^{3/2} edges containing no triangulation of the sphere. Also, we discuss several related results.

**Classif.: ** * 05C10 Topological graph theory

57M20 Two-dimensional complexes

05C35 Extremal problems (graph theory)

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