Skip to main content
SpringerLink
Log in

On Some Generalizations of the Property B Problem of an n-Uniform Hypergraph

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The extremal problem of hypergraph colorings related to the Erdős–Hajnal property B-problem is considered. Let k be a natural number. The problem is to find the value of mk(n) equal to the minimal number of edges in an n-uniform hypergraph that does not admit 2-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each color. In this paper, we obtain new lower bounds for mk(n).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Akolzin and D. Shabanov, “Colorings of hypergraphs with large number of colors,” Discrete Math., 339, No. 12, 3020–3031 (2016).

    Article  MathSciNet  Google Scholar 

  2. D. Cherkashin and J. Kozik, “A note on random greedy coloring of uniform hypergraphs,” Random Struct. Algorithms, 47, No. 3, 407–413 (2015).

    Article  MathSciNet  Google Scholar 

  3. Yu. A. Demidovich and A. M. Raigorodskii, “2-colorings of uniform hypergraphs,” Math. Notes, 100, No. 4, 629–632 (2016).

    Article  MathSciNet  Google Scholar 

  4. P. Erdős, “On a combinatorial problem,” Nordisk Mat. Tidskr., 11, 5–10 (1963).

    MathSciNet  MATH  Google Scholar 

  5. P. Erdős, “On a combinatorial problem. II,” Acta Math. Acad. Sci. Hungar., 15, No. 3-4, 445–447 (1964).

    Article  MathSciNet  Google Scholar 

  6. P. Erdős and A. Hajnal, “On a property of families of sets,” Acta Math. Acad. Sci. Hungar., 12, No. 1-2, 87–123 (1961).

    Article  MathSciNet  Google Scholar 

  7. P. Erdős and L. Lovász, “Problems and results on 3-chromatic hypergraphs and some related questions,” in: A. Hajnal, ed., Infinite and Finite Sets, Vol. 2, Colloq. Math. Soc. Jänos Bolyai, Vol. 10, North-Holland, Amsterdam (1975), pp. 609–627.

  8. A. E. Khuzieva and D. A. Shabanov, “On regular hypergraphs with high girth and high chromatic number,” Discrete Math. Appl., 25, No. 5, 277–294 (2015).

    Article  MathSciNet  Google Scholar 

  9. A. E. Khuzieva and D. A. Shabanov, “Quantitative estimates of characteristics for hypergraphs of large girth and large chromatic number,” Math. Notes, 98, No. 6, 995–998 (2015).

    Article  MathSciNet  Google Scholar 

  10. A. V. Kostochka, “Color-critical graphs and hypergraphs with few edges: A survey,” in: More Sets, Graphs and Numbers, Bolyai Soc. Math. Stud., Vol. 15, Springer, Berlin (2006), pp. 175–197.

  11. A. V. Kostochka and M. Kumbhat, “Coloring uniform hypergraphs with few edges,” Random Struct. Algorithms, 35, No. 3, 348–368 (2009).

    Article  MathSciNet  Google Scholar 

  12. J. Kozik and D. Shabanov, “Improved algorithms for colorings of simple hypergraphs and applications,” J. Combin. Theory. Ser. B., 116, 312–332 (2016).

    Article  MathSciNet  Google Scholar 

  13. A. Kupavskii and D. Shabanov, “Colorings of partial Steiner systems and their applications,” J. Math. Sci., 206, No. 5, 511–538 (2015).

    Article  MathSciNet  Google Scholar 

  14. A. Pluhár, “Greedy colorings for uniform hypergraphs,” Random Struct. Algorithms, 35, No. 2, 216–221 (2009).

    Article  MathSciNet  Google Scholar 

  15. J. Radhakrishnan and A. Srinivasan, “Improved bounds and algorithms for hypergraph 2-coloring,” Random Struct. Algorithms, 16, No. 1, 4–32 (2000).

    Article  MathSciNet  Google Scholar 

  16. A. M. Raigorodskii and D. A. Shabanov, “The Erdős–Hajnal problem of hypergraph colouring, its generalizations, and related problems,” Russ. Math. Surv., 66, No. 5, 933–1002 (2011).

  17. A. P. Rozovskaya, “On general two-colorings of uniform hypergraphs,” Dokl. Math., 80, No. 3, 837–839 (2009).

    Article  MathSciNet  Google Scholar 

  18. A. P. Rozovskaya, “Combinatorial extremum problems for 2-colorings of hypergraphs,” Math. Notes, 90, No. 4, 571–583 (2011).

    Article  MathSciNet  Google Scholar 

  19. D. A. Shabanov, “On a combinatorial problem of Erdős,” Dokl. Math., 69, No. 3, 359–362 (2004).

    MATH  Google Scholar 

  20. D. A. Shabanov, “Extremal problems for colourings of uniform hypergraphs,” Izv. Math., 71, No. 6, 1253–1290 (2007).

    Article  MathSciNet  Google Scholar 

  21. D. A. Shabanov, “Randomized algorithms for colourings of hypergraphs,” Sb. Math., 199, No. 7, 1089–1110 (2008).

    Article  MathSciNet  Google Scholar 

  22. D. Shabanov, “Coloring non-uniform hypergraphs without short cycles,” Graphs Combin., 30, No. 5, 1249–1260 (2014).

    Article  MathSciNet  Google Scholar 

  23. D. Shabanov, “Around Erdős–Lovász problem on colorings of non-uniform hypergraphs,” Discrete Math., 338, No. 11, 1976–1981 (2015).

  24. D. Shabanov, “Equitable two-colorings of uniform hypergraphs,” European J. Combin., 43, 185–203 (2015).

    Article  MathSciNet  Google Scholar 

  25. Z. Szabó, “An application of Lovász local lemma — a new lower bound for the van der Waerden number,” Random Struct. Algorithms, 1, No. 3, 343–360 (1990).

    Article  Google Scholar 

  26. S. M. Teplyakov, “Upper bound in the P. Erdős–Hajnal problem of hypergraph coloring,” Math. Notes, 93, No. 1, 191–195 (2013).

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Moscow, Russia

    Yu. A. Demidovich

Authors
  1. Yu. A. Demidovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. A. Demidovich.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 23, No. 1, pp. 95–122, 2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Demidovich, Y.A. On Some Generalizations of the Property B Problem of an n-Uniform Hypergraph. J Math Sci 262, 457–475 (2022). https://doi.org/10.1007/s10958-022-05828-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-05828-6

Search

Navigation