Abstract
Confirming a conjecture of Vera T. Sós in a very strong sense, we give a complete solution to Turán's hypergraph problem for the Fano plane. That is we prove for n≥8 that among all 3-uniform hypergraphs on n vertices not containing the Fano plane there is indeed exactly one whose number of edges is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for n = 7 there is exactly one other extremal configuration with the same number of edges: the hypergraph arising from a clique of order 7 by removing all five edges containing a fixed pair of vertices.
For sufficiently large values n this was proved earlier by Füredi and Simonovits, and by Keevash and Sudakov, who utilised the stability method.
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References
B. Bollobás: Extremal graph theory, Dover Publications, Inc., Mineola, NY, 2004.
W. G. Brown,: On an open problem of Paul Turán concerning 3-graphs, Studies in pure mathematics, Birkhäuser, Basel (1983), p91–93.
D. De Caen and Z. Füredi: The maximum size of 3-uniform hypergraphs not containing a Fano plane, . Combin. Theory Ser. B78 (2000), 274–276.
P. Erdős: Paul Turán, 1910-1976: his work in graph theory, J. Graph Theory1, (1977), 97–101.
P. Erdős, P. A. Hajnal, A., V. T. Sós and E. Szemerédi: More results on Ramsey-Turan type problems, Combinatorica 3 (1983), 69–81.
Z. Füredi: Personal communication.
Z. Füredi and A. Kündgen: Turán problems for integer-weighted graphs, J. Graph Theory40 (2002), 195–225.
Z. Füredi and M. Simonovits: Triple systems not containing a Fano configuration, Combin. Probab. Comput.14 (2005), 467–484.
Gy. Katona, T. Nemetz and M. Simonovits: On a problem of Turán in the theory of graphs, Mat. Lapok15 (1964), 228–238. (In Hungarian, with Russian and English summaries.)
P. Keevash and D. Mubayi: The Turán number of F3,3, Combin. Probab. Comput.21 (2012), 451–456.
P. Keevash and B. Sudakov: The Turán number of the Fano plane, Combinatorica25 (2005), 561–574.
A. V. Kostochka: A class of constructions for Turán's (3,4)-problem, Combinatorica2 (1982), 187–192.
C. M. Lüders and Ch. Reiher: The Ramsey-Turan problem for cliques, Israel Journal of Mathematics230 (2019), 613–652.
C. M. Lüders and Ch. Reiher: Weighted variants of the Andrasfai-Erdős-Sós Theorem, to appear in Journal of Combinatorics.
D. Mubayi and V. Rödl: On the Turán number of triple systems, J. Combin. Theory Ser. A100 (2002), 136–152.
M. Pasch: Vorlesungenüber neuere Geometrie, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], second edition, B. G. Teubner, Leipzig und Berlin (1912). (In German).
A. A. Razborov: Flag algebras, J. Symbolic Logic72 (2007), 1239–1282.
A. A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math.24 (2010), 946–963.
V. Rödl and A. Sidorenko: On the jumping constant conjecture for multigraphs, J. Combin. Theory Ser. A69 (1995), 347–357.
M. Simonovits: A method for solving extremal problems in graph theory, stability problems, Theory of Graphs, Proc. Colloq., Tihany, 1966, Academic Press, New York, 279–319, 1968.
V. T. Sós: Remarks on the connection of graph theory, finite geometry and block designs, Colloquio Internazionale sulle Teorie Combinatorie, Roma, 1973, Accad. Naz. Lincei, Rome, 223–233, 1976.
P. Turán: Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok48 (1941), 436–452. (In Hungarian, with German summary.)
W. T. Tutte: The factorization of linear graphs, J. London Math. Soc.22 (1947), 107–111.
Acknowledgement
We would like to thank Miklós Simonovits for sending us a copy of [21], Zoltán Füredi [6] for further information regarding the history of the problem, and the referees for a careful reading of this article.
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The second author was supported by the European Research Council (ERC grant PEPCo 724903).