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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.

##### 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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##### References:
 [1] W. G. Brown, On graphs that do not contain a Thomsen graph,Canad. Math. Bull. 9 (1966) 281–285. · Zbl 0178.27302 [2] P. Erdos andA. H. Stone, On the structure of linear graphs,Bull. Amer. Math. Soc. 52 (1946), 1087–1091. · Zbl 0063.01277 [3] P. Erdos andT. Gallai, On maximal paths and circuits of graphs,Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356. · Zbl 0090.39401 [4] P. Erdos, Extremal problems in graph theory,Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Prague, 1964, 29–36. [5] P. Erdos, On extremal problems of graphs and generalized graphs,Israel J. Math. 2 (1964), 183–190. · Zbl 0129.39905 [6] P. Erdos, A. Rényi andV. T. Sós, On a problem of graph theory,Studia Sci. Math. Hungar. 1 (1966), 215–235. [7] P. Erdos andM. Simonovits, A limit theorem in graph theory,Studia Sci. Math. Hungar. 1 (1966), 51–57. [8] P. Erdos andM. Simonovits, Some extremal problems in graph theory,Combinatorial Theory and its Applications (Colloq. Math. J. Bolyai4), Amsterdam-London, 1970, 377–390. [9] P. Erdos andD. J. Kleitman, On coloring graphs to maximize the proportion of multicolored k-edges,J. Combinatorial Theory 5 (1968) 164–169. · Zbl 0167.22302 [10] P. Erdos andV. T. Sós, Some remarks on Ramsey’s and Turán’s theorem,Combinatorial Theory and its Applications (Colloq. Math. J. Bolyai4), Amsterdam-London, 1970, 395–401. [11] P. Erdos, On some extremal properties onr-graphs,Discrete Math. 1 (1971), 1–6. · Zbl 0211.27003 [12] M. K. Fort Jr. andG. A. Hedlund, Minimal coverings of pairs by triples,Pacific J. Math. 8 (1958) 709–719. · Zbl 0084.01401 [13] F. Harary,Graph Theory, Reading, Mass., 1969. [14] Gy. Katona, T. Nemetz andM. Simonovits, Újabb bizonyítás a Turán-féle gráftételre és megjegyzések bizonyos általánosításaira,Mat. Lapok 15 (1964), 228–238. [15] T. Kovári, V. T. Sós andP. Turán, On a problem of K. Zarankiewicz,Colloq. Math. 3 (1955), 50–57. [16] G. Ringel, Extremal problems in the theory of graphs,Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Prague, 1964, 85–90. [17] H. Ryser,Combinatorial Mathematics, New York, 1963. · Zbl 0112.24806 [18] M. Simonovits, Extremal graph problems with conditions,Combinatorial Theory and its Appl. (Colloq. Math. J. Bolyai4), Amsterdam-London, 1970, 999–1011. [19] J. Singer, A theorem in finite projective geometry and some applications to number theory,Trans. Amer. Math. Soc. 43 (1938), 377–385. · Zbl 0019.00502 [20] P. Turán, Egy gráfelméleti szélsoérték-feladatról,Mat. Fiz. Lapok 48 (1941), 436–452. [21] P. Turán, On the theory of graphs,Colloq. Math. 3 (1954), 19–30. · Zbl 0055.17004
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