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Induced Turán Numbers


The classical Kővári–Sós–Turán theorem states that if G is an n-vertex graph with no copy of K s,t as a subgraph, then the number of edges in G is at most O(n 2−1/s ). We prove that if one forbids K s,t as an induced subgraph, and also forbids any fixed graph H as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in a K r -free graph with no induced copy of K s,t . This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

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[1] Alon, N. and Shikhelman, C. (2016) Many T copies in H-free graphs. J. Combin. Theory Ser. B 121 146172.
[2] Balogh, J., Bollobás, B. and Weinreich, D. (2000) The speed of hereditary properties of graphs. J. Combin. Theory Ser. B 79 131156.
[3] Bermond, J., Bond, J., Paoli, M. and Peyrat, C. (1983) Graphs and interconnection networks: Diameter and vulnerability. In Surveys in Combinatorics: Proceedings of the Ninth British Combinatorics Conference, Vol. 82 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 1–30.
[4] Boehnlein, E. and Jiang, T. (2012) Set families with a forbidden induced subposet. Combin. Probab. Comput. 21 496511.
[5] Bollobás, B. and Győri, E. (2008) Pentagons vs. triangles. Discrete Math. 308 43324336.
[6] Bollobás, B. and Thomason, A. (1997) Hereditary and monotone properties of graphs. In The Mathematics of Paul Erdős II (Graham, R. L. et al., eds), pp. 70–78.
[7] Chudnovsky, M. (2014) The Erdős–Hajnal conjecture: A survey. J. Graph Theory 75 178190.
[8] Chung, F. R. K., Gyárfás, A., Tuza, Z. and Trotter, W. T. (1990) The maximum number of edges in 2K 2-free graphs of bounded degree. Discrete Math. 81 129135.
[9] Chung, M., Jiang, T. and West, D. Induced Turán problems: Largest P m -free graphs with bounded degree. Submitted.
[10] Chung, M. and West, D. (1993) Large P 4-free graphs with bounded degree. J. Graph Theory 17 109116.
[11] Chung, M. and West, D. (1996) Large 2P 3-free graphs with bounded degree. Discrete Math. 150 6979.
[12] Conlon, D. (2012) On the Ramsey multiplicity of complete graphs. Combinatorica 32 171186.
[13] Erdős, P. (1962) On the number of complete subgraphs contained in certain graphs. Publ. Math. Inst. Hung. Acad. Sci., VII, Ser. A 3 459464.
[14] Erdős, P. (1964) On extremal problems of graphs and generalized graphs. Israel J. Math. 2 183190.
[15] Erdős, P., Frankl, P. and Rödl, V. (1986) The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin. 2 113121.
[16] Erdős, P. and Hajnal, A. (1989) Ramsey-type theorems. Discrete Appl. Math. 25 3752.
[17] Erdős, P. and Simonovits, M. (1982) Compactness results in extremal graph theory. Combinatorica 2 275288.
[18] Erdős, P. and Stone, A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52 10871091.
[19] Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Comput. Math. 2 463470.
[20] Fisher, D. and Ryan, J. (1992) Bounds on the number of complete subgraphs. Discrete Math. 103 313320.
[21] Fox, J. and Sudakov, B. (2011) Dependent random choice. Random Struct. Alg. 38 6899.
[22] Füredi, Z. (1996) New asymptotics for bipartite Turán numbers. J. Combin. Theory Ser. A 75 141144.
[23] 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.
[24] Gyárfás, A., Hubenko, A. and Solymosi, J. (2002) Large cliques in C 4-free graphs. Combinatorica 22 269274.
[25] Győri, E. and Li, H. (2012) The maximum number of triangles in C 2k + 1-free graphs. Combin. Probab. Comput. 21 187191.
[26] Hatami, H., Hladký, J., Král', D., Norine, S. and Razborov, A. (2013) On the number of pentagons in triangle-free graphs. J. Combin. Theory Ser. A 120 722732.
[27] Li, Y., Rousseau, C. and Zang, W. (2001) Asymptotic upper bounds for Ramsey functions. Graphs Combin. 17 123128.
[28] Lu, L. and Milans, K. (2015) Set families with forbidden subposets. J. Combin. Theory Ser. A 136 126142.
[29] Parsons, T. D. (1973) The Ramsey numbers r(P m , K n ). Discrete Math. 6 159162.
[30] Prömel, H. and Steger, A. (1991) Excluding induced subgraphs I: Quadrilaterals. Random Struct. Alg. 2 5571.
[31] Prömel, H. and Steger, A. (1993) Excluding induced subgraphs II: Extremal graphs. Discrete Appl. Math. 44 283294.
[32] Prömel, H. and Steger, A. (1992) Excluding induced subgraphs III: A general asymptotic. Random Struct. Alg. 3 1931.
[33] Razborov, A. (2010) On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. 24 946963.
[34] Sidorenko, A. (1995) What we know and what we do not know about Turán numbers. Graphs Combin. 11 179199.
[35] Simonovits, M. and Sós, V. T. (2001) Ramsey–Turán theory. Discrete Math. 229 293340.
[36] Sós, V. T. (1969) On extremal problems in graph theory. In Proceedings of the Calgary International Conference on Combinatorial Structures and their Application, Gordon and Breach, NY, pp. 407–410.
[37] Turán, P. (1941) On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok 48 436452.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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