1995
DOI: 10.1007/bf01929486
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What we know and what we do not know about Turán numbers

Abstract: Abstract. The numbers which are traditionally named in honor of Paul Turfin were introduced by him as a generalization of a problem he solved in 1941. The general problem of Tur~in having an extremely simple formulation but being extremely hard to solve, has become one of the most fascinating extremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless. The Definition and Equivalent FormulationsA system of r-element subsets (blocks) of an n-element set Xn… Show more

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Cited by 155 publications
(135 citation statements)
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“…He also gave upper bounds and conjectures for T (n, 4, 3) and T (n, 5, 3), which stimulated much of the research. The results labeled 'Turán theory' in our tables either are described in recent survey papers by de Caen [8] and Sidorenko [29], or follow from constructions due to de Caen, Kreher, and Wiseman [10] or to Sidorenko [28].…”
Section: Turán Theorymentioning
confidence: 79%
See 1 more Smart Citation
“…He also gave upper bounds and conjectures for T (n, 4, 3) and T (n, 5, 3), which stimulated much of the research. The results labeled 'Turán theory' in our tables either are described in recent survey papers by de Caen [8] and Sidorenko [29], or follow from constructions due to de Caen, Kreher, and Wiseman [10] or to Sidorenko [28].…”
Section: Turán Theorymentioning
confidence: 79%
“…For the rest: If t = 2, the lower bound is explained by Mills and Mullin [19] when it is less than 14 or has v ≤ 5, or explained by Todorov [34] [29].…”
Section: Tables Of Upper Bounds On C(v K T)mentioning
confidence: 99%
“…In particular, when e = v r we get the well known Turán problem of determining the maximum possible number of edges in an r-graph that contains no complete r-graph on v vertices. See the surveys [8], [11], [14], and [21] for results and references on this and other graph and hypergraph Turán problems. In 1973, Brown, Erdős and Sós [5], [6] initiated the study of the function f for r-graphs (r ≥ 3).…”
Section: Introductionmentioning
confidence: 99%
“…By choice of partition there are at least as many good edges containing a as bad. From (9) we see that a is in at least ways. Once X and Y have been chosen, to make E good b is constrained to lie in some particular class V i of the partition, so can be chosen in at most 1 2 + 1 10 η n ways.…”
mentioning
confidence: 99%
“…numbers ex(n, F) when r > 2 is notoriously difficult, and exact results on hypergraph Turán numbers are very rare (see [3,9] for surveys). In this paper we obtain such a result for a sequence of hypergraphs introduced by Frankl. Let C (2k) r be the 2k-uniform hypergraph obtained by letting P 1 , · · · , P r be pairwise disjoint sets of size k and taking as edges all sets P i ∪ P j with i = j.…”
mentioning
confidence: 99%