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Supersaturation of even linear cycles in linear hypergraphs

  • Tao Jiang (a1) and Liana Yepremyan (a2)


A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n-vertex graph not containing C2k, the cycle of length 2k, is O(n1+1/k). Simonovits established a corresponding supersaturation result for C2k’s, showing that there exist positive constants C,c depending only on K such that every n-vertex graph G with e(G) ≥ Cn1+1/k contains at least c(e(G)/v(G))2k copies of C2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant).

In this paper we extend Simonovits’ result to a supersaturation result of r-uniform linear cycles of even length in r-uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.


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Research supported in part by National Science Foundation grant DMS-1400249 and DMS-1855542. Email:

Research supported by Marie Sklodowska Curie Global Fellowship 846304.



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[1]Balogh, J., Narayanan, B. and Skokan, J. (2019) The number of hypergraphs without linear cycles. J. Combin. Theory Ser. B 134 309321.
[2]Brown, W. G., Erdős, P. and Sós, V. (1973) On the existence of triangulated spheres in 3-graphs and related problems. Period. Math. Hungar. 3 221228.
[3]Bondy, A. and Simonovits, M. (1974) Cycles of even length in graphs. J. Combin. Theory Ser. B 16 97105.
[4]Bukh, B. and Jiang, Z. (2017) A bound on the number of edges in graphs without an even cycle. Combin. Probab. Comput. 26 115.
[5]Collier-Cartaino, C., Graber, N. and Jiang, T. (2018) Linear Turán numbers of linear cycles and cycle-complete Ramsey numbers. Combin. Probab. Comput. 27 358386.
[6]Erdős, P. and Simonovits, M. (1966) A limit theorem in graph theory. Studia Sci. Math. Hungar. 1 5157.
[7]Erdős, P. and Simonovits, M. (1970) Some extremal problems in graph theory. In Combinatorial Theory and its Applications I (Proc. Colloq. Balatonfüred, 1969), North-Holland, pp. 377390.
[8]Erdős, P. and Simonovits, M. (1983) Supersaturated graphs and hypergraphs. Combinatorica 3 181192.
[9]Erdős, P. and Simonovits, M. (1984) Cube-supersaturated graphs and related problems. In Progress in Graph Theory (Waterloo, Ont., 1982), Academic Press, pp. 203218.
[10]Erdős, P. and Stone, H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 10871091.
[11]Ergemlidze, B., Győri, E. and Methuku, A. (2019) Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs. J. Combin. Theory Ser. A 163 163181.
[12]Faudree, R. and Simonovits, M. (1983) On a class of degenerate extremal graph problems. Combinatorica 3 8393.
[13]Faudree, R. and Simonovits, M. Cycle-supersaturated graphs. In preparation.
[14]Füredi, Z. and Jiang, T. (2014) Hypergraph Turán numbers of linear cycles. J. Combin. Theory Ser. A 123 252270.
[15]Füredi, Z. and Simonovits, M. (2013) The history of the degenerate (bipartite) extremal graph problems. In Erdős Centennial, Vol. 25 of Bolyai Society Mathematical Studies, Springer, pp. 169264. See also arXiv:1306.5167.
[16]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.
[17]Keevash, P. (2011) Hypergraph Turán problems. In Surveys in Combinatorics, Cambridge University Press, pp. 83140.
[18]Keevash, P. (2014) The existence of designs. arXiv:1401.3665.
[19]Kostochka, A., Mubayi, D. and Verstraëte, J. (2015) Turán problems and shadows I: Paths and cycles. J. Combin. Theory Ser. A 129 5779.
[20]Morris, R. and Saxton, D. (2016) The number of C2l-free graphs. Adv. Math. 298 534580.
[21]Pikhurko, O. (2012) A note on the Turán function of even cycles. Proc. Amer. Math. Soc. 140 36873692.
[22]Rödl, V. (1985) On a packing and covering problem. Europ. J. Combin. 6 6978.
[23]Ruzsa, I. and Szemerédi, E. Triples systems with no six points carrying three triangles. In Combinatorics II (Keszthely 1976), Vol. 18 of Colloquia Mathematica Societatis János Bolyai, pp. 939945.
[24]Sidorenko, A. (1991) Inequalities for functionals generated by bipartite graphs (in Russian). Diskret. Mat. 3 50–65. English translation: Discrete Math. Appl. 2 (1992) 489504.
[25]Sidorenko, A. (1993) A correlation inequality for bipartite graphs. Graphs Combin. 9 201204.
[26]Simonovits, M. (1982) Extremal graph problems, degenerate extremal problems, and supersaturated graphs. In Progress in Graph Theory (Waterloo 1982), Academic Press, pp. 419437.
[27]Verstraëte, J. (2000) On arithmetic progressions of cycle lengths in graphs. Combin. Probab. Comput. 9 369373.
[28]Wilson, R. (1972) An existence theory for pairwise balanced designs I: Composition theorems and morphisms. J. Combin. Theory Ser. A 13 220245.
[29]Wilson, R. (1972) An existence theory for pairwise balanced designs II: The structure of PBD-closed sets and the existence conjectures. J. Combin. Theory Ser. A 13 246273.
[30]Wilson, R. (1975) An existence theory for pairwise balanced designs III: Proof of the existence conjectures. J. Combin. Theory Ser. A 18 7179.

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Supersaturation of even linear cycles in linear hypergraphs

  • Tao Jiang (a1) and Liana Yepremyan (a2)


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