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Extremal problems for triple systems. (English) Zbl 0817.05015
Summary: Turán-type problems for several triple systems arising from $$(k,k- 2)$$- configurations [i.e. $$(k- 2)$$ triples on $$k$$ vertices] are considered. It is shown that every Steiner triple system contains a $$(k,k- 2)$$- configuration for some $$k< c\log n/\log\log n$$. Moreover, the Turán numbers of $$(k,k- 2)$$-trees are determined asymptotically to be $$((k- 3)/3)\cdot(\begin{smallmatrix} n\\ 2\end{smallmatrix})(1- o(1))$$. Finally, anti- Pasch hypergraphs avoiding $$(5,3)$$- and $$(6,4)$$-configurations are considered.

##### MSC:
 05B07 Triple systems 51E10 Steiner systems in finite geometry 05C65 Hypergraphs
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##### References:
 [1] , and , Design theory, Cambridge University Press, Cambridge, 1986. [2] Bose, Ann. Eugenics 9 pp 353– (1939) · Zbl 0023.00102 [3] ”On a method of constructing Steiner triple systems,” Contributions to probability and statistics, Stanford Univ. Press, Stanford, CA, 1960, pp. 133–141. [4] Steiner triple systems without forbidden subconfigurations, Mathematisch Centrum Amsterdam, ZW 104/77. [5] Brown, Canad. Math. Bull. 9 pp 281– (1966) · Zbl 0178.27302 [6] , and , ”Some extremal problems on r-graphs,” in New directions in the theory of graphs, Proc. 3rd Ann Arbor Conf. on Graph Theory, Univ. Michigan, Academic Press, New York, 1971, pp. 53–63. [7] Colbourn, Australasian J. Combinatorics 4 pp 143– (1991) [8] Erdös, Izvestiya Naustno-Issl. Inst. Mat. i Meh. Tomsk 2 pp 74– (1938) [9] Erdös, Israel Journal of Mathematics 2 pp 183– (1964) [10] Erdös, Studia Sci. Math. Hungar. 1 pp 215– (1966) [11] Robinson, Math. Comput. 29 pp 223– (1975) [12] Rödl, J. Comb. 5 pp 69– (1985) · Zbl 0565.05016 [13] Sós, Periodica Mathematica Hungarica 3 pp 221– (1973)
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