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Hypergraph Based Berge Hypergraphs

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Abstract

Fix a hypergraph \({\mathcal {F}}\). A hypergraph \({\mathcal {H}}\) is called a Berge copy of \({\mathcal {F}}\) or Berge-\({\mathcal {F}}\) if we can choose a subset of each hyperedge of \({\mathcal {H}}\) to obtain a copy of \({\mathcal {F}}\). A hypergraph \({\mathcal {H}}\) is Berge-\({\mathcal {F}}\)-free if it does not contain a subhypergraph which is Berge copy of \({\mathcal {F}}\). This is a generalization of the usual, graph-based Berge hypergraphs, where \(\mathcal {F}\) is a graph. In this paper, we study extremal properties of hypergraph based Berge hypergraphs and generalize several results from the graph-based setting. In particular, we show that for any r-uniform hypergraph \({\mathcal {F}}\), the sum of the sizes of the hyperedges of a (not necessarily uniform) Berge-\({\mathcal {F}}\)-free hypergraph \({\mathcal {H}}\) on n vertices is \(o(n^r)\) when all the hyperedges of \({\mathcal {H}}\) are large enough. We also give a connection between hypergraph based Berge hypergraphs and generalized hypergraph Turán problems.

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References

  1. Alon, N., Shikhelman, C.: Many \(T\) copies in \(H\)-free graphs. J. Combin. Theory Ser. B 121, 146–172 (2016)

    Article  MathSciNet  Google Scholar 

  2. Alon, N., Shapira, A.: On an extremal hypergraph problem of Brown. Erdős and Sós. Combinatorica 26, 627–645 (2006)

    Article  MathSciNet  Google Scholar 

  3. Anstee, R., Salazar, S.: Forbidden Berge hypergraphs. Electron. J. Combin. 24(1), 59 (2017)

    Article  MathSciNet  Google Scholar 

  4. Axenovich, M., Gyárfás, A.: A note on Ramsey numbers for Berge-G hypergraphs. Discret. Math. 342, 1245–1252 (2019)

    Article  MathSciNet  Google Scholar 

  5. Berge, C.: Graphs and Hypergraphs. North-Holland Pub, Amsterdam (1973)

    MATH  Google Scholar 

  6. de Caen, D.: Extension of a theorem of Moon and Moser on complete subgraphs. Ars Combin. 16, 5–10 (1983)

    MathSciNet  MATH  Google Scholar 

  7. English, S., Gerbner, D., Methuku, A., Palmer, C.: On the weight of Berge-\(F\)-free hypergraphs. Electron. J. Combin. 26(4), 7 (2019)

    Article  MathSciNet  Google Scholar 

  8. English, S., Gerbner, D., Methuku, A., Tait, M.: Linearity of saturation for Berge hypergraphs. Eur. J. Combin. 78, 205–213 (2019)

    Article  MathSciNet  Google Scholar 

  9. English, S., Graber, N., Kirkpatrick, P., Methuku, A., Sullivan, E.C.: Saturation of Berge hypergraphs. Discrete Math. 342, 1738–1761 (2019)

    Article  MathSciNet  Google Scholar 

  10. Füredi, Z., Kostochka, A., Luo, R.: Avoiding long Berge cycles. J. Combin. Theory Ser. B 137, 55–64 (2019)

    Article  MathSciNet  Google Scholar 

  11. Gerbner, D.: A note on the Turán number of a Berge odd cycle. Aust. J. Combin. 79, 205–214 (2021)

    MATH  Google Scholar 

  12. Gerbner, D., Methuku, A., Omidi, G., Vizer, M.: Ramsey problems for Berge hypergraphs. SIAM J. Discret. Math. 34, 351–369 (2020)

    Article  MathSciNet  Google Scholar 

  13. Gerbner, D., Methuku, A., Palmer, C.: General lemmas for Berge-Turán hypergraph problems. Eur. J. Combin. 86, Article 103082 (2020)

  14. Gerbner, D., Methuku, A., Vizer, M.: Asymptotics for the Turán number of Berge-\(K_{2, t}\). J. Combin. Theory Ser. B 137, 264–290 (2019)

    Article  MathSciNet  Google Scholar 

  15. Gerbner, D., Palmer, C.: Extremal results for Berge hypergraphs. SIAM J. Discrete. Math. 31, 2314–2327 (2017)

    Article  MathSciNet  Google Scholar 

  16. Gerbner, D., Palmer, C.: Counting copies of a fixed subgraph in \(F\)-free graphs. Eur. J. Combin. 82, Article 103001 (2019)

  17. Gerbner, D., Patkós, B.: Extremal Finite Set Theory, 1st Edition, CRC Press, New York (2018)

  18. Gowers, W.: Hypergraph regularity and the multidimensional Szemerédi theorem. Ann. Math. 166, 897–946 (2007)

    Article  MathSciNet  Google Scholar 

  19. Grósz, D., Methuku, A., Tompkins, C.: Uniformity thresholds for the asymptotic size of extremal Berge-\(F\)-free hypergraphs. Eur. J. Combin. 88, Article 103109 (2020)

  20. Győri, E.: Triangle-free hypergraphs. Comb. Probab. Comput. 15, 185–191 (2006)

    Article  MathSciNet  Google Scholar 

  21. Győri, E., Katona, G.Y., Lemons, N.: Hypergraph extensions of the Erdős-Gallai theorem. Eur. J. Comb. 58, 238–246 (2016)

    Article  Google Scholar 

  22. Győri, E., Lemons, N.: Hypergraphs with no cycle of a given length. Comb. Probab. Comput. 21, 193–201 (2012)

    Article  MathSciNet  Google Scholar 

  23. Kang, L., Liu, L., Lu, L., Wang, Z.: The extremal \(p\)-spectral radius of Berge-hypergraphs. Linear Algebra Appl. 610, 608–624 (2021)

    Article  MathSciNet  Google Scholar 

  24. Lazebnik, F., Verstraëte, J.: On hypergraphs of girth five. Electron. J. Comb. 10, R25 (2003)

    Article  MathSciNet  Google Scholar 

  25. Ma, J., Yuan, X., Zhang, M.: Some extremal results on complete degenerate hypergraphs. J. Combin. Theory Ser. A 154, 598–609 (2018)

    Article  MathSciNet  Google Scholar 

  26. Mubayi, D., Verstraëte, J.: A survey of Turán problems for expansions. Recent Trends Combin. pp. 117–143 (2016)

  27. Palmer, C., Tait, M., Timmons, C., Wagner, A.Z.: Turán numbers for Berge-hypergraphs and related extremal problems. Discrete. Math. 342, 1553–1563 (2019)

    Article  MathSciNet  Google Scholar 

  28. Pikhurko, O.: Exact computation of the hypergraph Turán function for expanded complete \(2\)-graphs. J. Combin. Theory Ser. B 103, 220–225 (2013)

    Article  MathSciNet  Google Scholar 

  29. Rödl, V., Nagle, B., Skokan, J., Schacht, M., Kohayakawa, Y.: From the cover: the hypergraph regularity method and its applications. Proc. Natl. Acad. Sci. 102, 8109–8113 (2005)

    Article  MathSciNet  Google Scholar 

  30. Salia, N., Tompkins, C., Wang, Z., Zamora, O.: Ramsey numbers of Berge-hypergraphs and related structures. Electron. J. Combin. 26(4), 40 (2019)

  31. Timmons, C.: On \(r\)-uniform linear hypergraphs with no Berge-\(K_{2, t}\). Electron. J. Combin. 24, 4.34 (2017)

  32. Xu, Z., Zhang, T., Ge, G.: Some extremal results on hypergraph Turán problems. arXiv preprint arXiv:1905.01685 (2019)

  33. Zykov, A.A.: On some properties of linear complexes. Matematicheskii sbornik 66, 163–188 (1949)

    MathSciNet  Google Scholar 

Download references

Funding

M. Balko was supported by the Grant No. 19-04113Y of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). D. Gerbner was supported in part by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and the National Research, Development and Innovation Office - NKFIH under the Grants K 116769, KH 130371 and SNN 12936. D. Kang was supported by the Institute for Basic Science, No. IBS-R029-C1, and the research leading to these results was also supported by the EPSRC, grant nos. EP/N019504/1. Y. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A6A3A04005963). C. Palmer was supported by a grant from the Simons Foundation #712036.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Martin Balko

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Dániel Gerbner

  3. School of Mathematics, University of Birmingham, Birmingham, UK

    Dong Yeap Kang

  4. Institute of Mathematical Sciences, Ewha Womans University, Seoul, South Korea

    Younjin Kim

  5. Department of Mathematical Sciences, University of Montana, Missoula, MT, 59812, USA

    Cory Palmer

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  1. Martin Balko
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  2. Dániel Gerbner
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  3. Dong Yeap Kang
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  4. Younjin Kim
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  5. Cory Palmer
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Correspondence to Younjin Kim.

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Balko, M., Gerbner, D., Kang, D.Y. et al. Hypergraph Based Berge Hypergraphs. Graphs and Combinatorics 38, 11 (2022). https://doi.org/10.1007/s00373-021-02419-1

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