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Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

  • CLAYTON COLLIER-CARTAINO (a1), NATHAN GRABER (a2) and TAO JIANG (a1)
Abstract

An r-uniform hypergraph is called an r-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear r-graph H and a positive integer n, the linear Turán number ex L (n,H) is the maximum number of edges in a linear r-graph G that does not contain H as a subgraph. For each ℓ ≥ 3, let C r denote the r-uniform linear cycle of length ℓ, which is an r-graph with edges e 1, . . ., e such that, for all i ∈ [ℓ−1], |e i e i+1|=1, |e e 1|=1 and e i e j = ∅ for all other pairs {i,j}, ij. For all r ≥ 3 and ℓ ≥ 3, we show that there exists a positive constant c = c r,ℓ, depending only r and ℓ, such that ex L (n,C r ) ≤ cn 1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.

Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constants a = a m,r and b = b m,r , depending only on m and r, such that \begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

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[1] Ajtai M., Komlós J. and Szemerédi E. (1980) A note on Ramsey numbers. J. Combin. Theory Ser. A 29 354360.
[2] Alon N. (1996) Independence numbers of locally sparse graphs and a Ramsey type problem. Random Struct. Alg. 9 271278.
[3] Alon N., Krivelevich M. and Sudakov B. (1999) Coloring graphs with sparse neighborhoods. J. Combin. Theory Ser. B 77 7382.
[4] Behrend F. (1946) On sets of integers which contain no three elements in arithmetic progression. Proc. Nat. Acad. Sci. 32 331332.
[5] Bohman T. and Keevash P. (2010) The early evolution of the H-free process. Invent. Math 181 291336.
[6] Bohman T. and Keevash P. Dynamic concentration of the triangle-free process. arXiv:1302.5963
[7] Bondy J. A. and Simonovits M. (1974) Cycles of even length in graphs. J. Combin. Theory Ser. B 16 97105.
[8] 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.
[9] Bukh B. and Jiang Z. (2017) A bound on the number of edges in graphs without an even cycle. Combin. Probab. Comput. 26 115.
[10] Caro Y. (1979) New results on the independence number. Technical Report, Tel Aviv University.
[11] Caro Y., Li Y., Rousseau C. and Zhang Y. (2000) Asymptotic bounds for some bipartite graph: Complete graph Ramsey numbers. Discrete Math. 220 5156.
[12] Das S., Lee C. and Sudakov B. (2013) Rainbow Turán problem for even cycles. Euro. J. Combin. 34 905915
[13] Erdős P., Faudree R., Rousseau C. and Schelp R. (1978) On cycle-complete graph Ramsey numbers. J. Graph Theory 2 5364.
[14] Erdős P. and Rado R. (1960) Intersection theorems for systems of sets. J. London Math Soc. (2) 35 8590.
[15] Erdős P. and Stone A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 10871091.
[16] Faudree R. and Simonovits M. (1983) On a class of degenerate extremal graph problems. Combinatorica 3 8393.
[17] Fiz Pontiveros G., Griffiths S. and Morris R. The triangle-free process and R(3,k). arXiv:1302.6279
[18] Füredi Z. (1996) On the number of edges of quadrilateral-free graphs. J. Combin. Theory Ser. B 68 16.
[19] Füredi Z. and Jiang T. (2014) Hypergraph Turán numbers of linear cycles. J. Combin. Theory Ser. A 123 252270.
[20] Füredi Z., Jiang T. and Seiver R. (2014) Exact solution of the hypergraph Turán problem for k-uniform linear paths. Combinatorica 34 299322.
[21] Füredi Z., Naor A. and Verstraëte J. (2006) On the Turán number for the hexagon. Adv. Math. 203 476496.
[22] Füredi Z. and Simonovits M. (2013) The history of degenerate (bipartite) extremal graph problems. In Erdős Centennial (Lovász L. et al., eds), Vol. 25 of Bolyai Society Mathematical Studies, Springer, pp. 169264.
[23] Győri E. and Lemons N. (2012) Hypergraphs with no cycle of a given length. Combin. Probab. Comput. 21 193201.
[24] Győri E. and Lemons N. (2012) 3-uniform hypergraphs avoiding a given odd cycle. Combinatorica 32 187203.
[25] Jiang T. and Seiver R. (2012) Turán numbers of subdivided graphs. SIAM J. Discrete Math. 26 12381255.
[26] Li Y. and Zang W. (2003) The independence number of graphs with a forbidden cycle and Ramsey numbers. J. Combin. Opt. 7 353359.
[27] Keevash P., Mubayi D., Sudakov B. and Verstraëte J. (2006) Rainbow Turán problems. Combin. Probab. Comput. 16 109126.
[28] Kim J. (1995) The Ramsey number R(3,t) has order of magnitude t 2/logt. Random Struct. Alg. 7 173207.
[29] Kostochka A., Mubayi D. and Verstraëte J. (2013) Hypergraph Ramsey numbers: Triangles versus cliques. J. Combin. Theory Ser. A 120 14911507.
[30] Kostochka A., Mubayi D. and Verstraëte J. Personal communications.
[31] Kostochka A., Mubayi D. and Verstraëte J. (2015) Turán problems and shadows I: Paths and cycles. J. Combin. Theory Ser. A 129 5779.
[32] Lazebnik F. and Verstraëte J. (2003) On hypergraphs of girth 5. Electron. J. Combin. 10 R25.
[33] Méroueh A. The Ramsey number of loose cycles versus cliques. arXiv:1504.03668
[34] Molloy M. and Reed B. (2002) Graph Colouring and the Probabilistic Method, Vol. 23 of Algorithms and Combinatorics, Springer.
[35] Pikhurko O. (2012) A note on the Turán function of even cycles. Proc. Amer. Math. Soc. 140 36873992.
[36] Roth K. F. (1951) On a problem of Heilbronn. J. London Math. Soc. 26 198204.
[37] Ruzsa I. (1993) Solving a linear equation in a set of integers I. Acta Arithmetica 65 259282.
[38] Ruzsa I. and Szemerédi E. (1978) Triple systems with no six points carrying three triangles. Colloq. Math. Soc. J. Bolyai 18 939945.
[39] Sudakov B. (2002) A note on odd cycle-complete graph Ramsey numbers. Electron. J. Combin. 9 N1.
[40] Verstraëte J. (2000) On arithmetic progressions of cycle lengths in graphs. Combin. Probab. Comput. 9 369373.
[41] Wei V. K. (1981) A lower bound on the stability number of a simple graph. Technical memorandum TM 81-11217-9, Bell Laboratories.
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Combinatorics, Probability and Computing
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