Skip to main content
SpringerLink
Log in

Systems of Overlap Representation for Families of Intervals

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Two sets overlap if they intersect and neither contains the other. Given a family \(\mathcal {F}\) of sets, a system of overlap representation (SOR) for \(\mathcal {F}\) assigns to each set \(X\in \mathcal {F}\) a subset \(S_X\) of X, called its representative set, so that the representative sets chosen for any two overlapping members of \(\mathcal {F}\) intersect. Let \(\mathcal {F}_n\) be the family of intervals of integers contained in \(\{1,\ldots ,n\}\), and let f(n) be the minimum of the maximum size of the sets in an SOR of \(\mathcal {F}_n\). We prove

$$(1-o(1))\lg (n-1)<f(n)\le 2\lg (n-1).$$

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

Availability of Data and Material

There are none used.

References

  1. Burton, J., Stapleton, G.: Special issue on Euler and Venn diagrams: guest editors’ introduction. J. Log. Lang. Inf. 24, 357–359 (2015)

    Article  MathSciNet  Google Scholar 

  2. Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26–30 (1935)

    Article  Google Scholar 

  3. Jones, M., Nikovski, D., Imamura, M., Hirata, T.: Exemplar learning for extremely efficient anomaly detection in real-valued time series. Data Min. Knowl. Discov. 30, 1427–1454 (2016)

    Article  MathSciNet  Google Scholar 

  4. Kürschák, J.: A Harmonikus Sorról. Mat. és Fiz. Lapok 27, 299–300 (1918)

    Google Scholar 

  5. Simonetto, P., Archambault, D.: Fully automatic visualisation of overlapping sets. Comput. Graph. Forum 28, 967–974 (2009)

    Article  Google Scholar 

  6. Theisinger, L.: Bemerkung über die harmonische Reihe. Monatsh. f. Mathematik und Physik 26, 132–134 (1915)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Keith Conrad for providing the references for the nonintegrality of harmonic numbers.

Funding

Funding sources are listed as footnotes on the first page of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Louisville, Louisville, USA

    Adam S. Jobson, André E. Kézdy & D. Jacob Wildstrom

  2. Department of Mathematics, West Virginia University, Morgantown, USA

    Kevin G. Milans

  3. Department of Mathematics, Zhejiang Normal University, Jinhua, China

    Douglas B. West

  4. Department of Mathematics, University of Illinois, Urbana, IL, USA

    Douglas B. West

Authors
  1. Adam S. Jobson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. André E. Kézdy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kevin G. Milans
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Douglas B. West
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. D. Jacob Wildstrom
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed (In mathematics, the universal convention is that all authors contributed equally to all aspects of the manuscript.)

Corresponding author

Correspondence to Douglas B. West.

Ethics declarations

Conflict of interest

There are none.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research supported by National Natural Science Foundation of China Grants NSFC 11871439, 11971439, and U20A2068.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jobson, A.S., Kézdy, A.E., Milans, K.G. et al. Systems of Overlap Representation for Families of Intervals. Graphs and Combinatorics 38, 36 (2022). https://doi.org/10.1007/s00373-021-02417-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-021-02417-3

Keywords

Mathematics Subject Classification

Search

Navigation