## Abstract

Let *Σ*
_{
k
} consist of all *k*-graphs with three edges *D*
_{1}, *D*
_{2}, *D*
_{3} such that |*D*
_{1} ∩ *D*
_{2}| = *k* − 1 and *D*
_{1} Δ *D*
_{2} ⊆ *D*
_{3}. The exact value of the Turán function ex(*n*, *Σ*
_{
k
}) was computed for *k* = 3 by Bollobás [Discrete Math. **8** (1974), 21–24] and for *k* = 4 by Sidorenko [Math Notes **41** (1987), 247–259].

Let the *k*-graph *T*
_{
k
} ∈ *Σ*
_{
k
} have edges

Frankl and Füredi [J. Combin. Theory Ser. (A) **52** (1989), 129–147] conjectured that there is *n*
_{0} = *n*
_{0}(*k*) such that ex(*n*, *T*
_{
k
}) = ex(*n*, *Σ*
_{
k
}) for all *n* ≥ *n*
_{0} and had previously proved this for *k* = 3 in [Combinatorica **3** (1983), 341–349]. Here we settle the case *k* = 4 of the conjecture.

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## References

- [1]
B. Bollobás: Three-graphs without two triples whose symmetric difference is contained in a third,

*Discrete Math.***8**(1974), 21–24. - [2]
D. de Caen: Uniform hypergraphs with no block containing the symmetric difference of any two other blocks,

*Congres. Numer.***47**(1985), 249–253. - [3]
G. Elek and B. Szegedy: Limits of hypergraphs, removal and regularity lemmas. A non-standard approach; submitted (2007), arXiv:0705.2179.

- [4]
P. Erdős and M. Simonovits: Supersaturated graphs and hypergraphs,

*Combinatorica***3(2)**(1983), 181–192. - [5]
P. Frankl and Z. Füredi: A new generalization of the Erdős-Ko-Rado theorem,

*Combinatorica***3(3–4)**(1983), 341–349. - [6]
P. Frankl and Z. Füredi: Extremal problems whose solutions are the blowups of the small Witt-designs,

*J. Combin. Theory Ser. (A)***52**(1989), 129–147. - [7]
Z. Füredi and M. Simonovits: Triple systems not containing a Fano configuration,

*Combin. Prob. Computing***14**(2005), 467–488. - [8]
W. T. Gowers: Hypergraph regularity and the multidimensional Szemerédi’s theorem, submitted (2007), arXiv:0710.3032.

- [9]
G. O. H. Katona: Extremal problems for hypergraphs,

*Combinatorics*vol.**56**, Math. Cent. Tracts, 1974, pp. 13–42. - [10]
G. O. H. Katona, T. Nemetz and M. Simonovits: On a graph problem of Turán (In Hungarian),

*Mat. Fiz. Lapok***15**(1964), 228–238. - [11]
P. Keevash and D. Mubayi: Stability results for cancellative hypergraphs,

*J. Combin. Theory Ser. (B)***92**(2004), 163–175. - [12]
P. Keevash and B. Sudakov: The Turán number of the Fano plane,

*Combinatorica***25(5)**(2005), 561–574. - [13]
P. Keevash and B. Sudakov: On a hypergraph Turán problem of Frankl,

*Combinatorica***25(6)**(2005), 673–706. - [14]
W. Mantel: Problem 28,

*Winkundige Opgaven***10**(1907), 60–61. - [15]
D. Mubayi and O. Pikhurko: A new generalization of Mantel’s theorem to

*k*-graphs,*J. Combin. Theory Ser. (B)***97**(2007), 669–678. - [16]
B. Nagle, V. Rödl and M. Schacht: The counting lemma for regular

*k*-uniform hypergraphs,*Random Struct. Algorithms***28**(2005), 113–179. - [17]
O. Pikhurko: Exact computation of the hypergraph Turán function for expanded complete 2-graphs, arXiv:math/0510227, accepted by

*J. Comb. Th. Ser. (B)*. The publication is suspended because of a disagreement over copyright, see http://www.math.cmu.edu/:_pikhurko/Copyright.html, 2005. - [18]
V. Rödl and J. Skokan: Regularity lemma for

*k*-uniform hypergraphs,*Random Struct. Algorithms***25**(2004), 1–42. - [19]
V. Rödl and J. Skokan: Applications of the regularity lemma for uniform hypergraphs,

*Random Struct. Algorithms***28**(2006), 180–194. - [20]
J. B. Shearer: A new construction for cancellative families of sets,

*Electronic J. Combin.***3**(1996), 3 pp. - [21]
A. F. Sidorenko: The maximal number of edges in a homogeneous hypergraph containing no prohibited subgraphs,

*Math Notes***41**(1987), 247–259, Translated from*Mat. Zametki*. - [22]
T. Tao: A variant of the hypergraph removal lemma,

*J. Combin. Theory Ser. (A)***113**(2006), 1257–1280. - [23]
P. Turán: On an extremal problem in graph theory (in Hungarian),

*Mat. Fiz. Lapok***48**(1941), 436–452.

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Reverts to public domain after 28 years from publication.

Partially supported by the National Science Foundation, Grant DMS-0457512.

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Pikhurko, O. An exact Turán result for the generalized triangle.
*Combinatorica* **28, **187–208 (2008). https://doi.org/10.1007/s00493-008-2187-2

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### Mathematics Subject Classification (2000)

- 05D05
- 05C35