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Progress on poset-free families of subsets

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

Increasing attention is being paid to the study of families of subsets of an n-set that contain no subposet P. Especially, we are interested in such families of maximum size given P and n. For certain P this problem is solved for general n, while for other P it is extremely challenging to find even an approximate solution for large n. It is conjectured that for any P, the maximum size is asymptotic to a constant times \(\binom{n}{\lfloor \frac{n} {2} \rfloor }\), where the constant is a certain integer depending on P. This survey has two purposes. First, we want to bring this exciting line of research to the attention of a wider audience. Second, we want to make experts aware of the broad range of recent progress in the area.

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Notes

  1. 1.

    After this survey went to press, we learned that Arès Méroueh announced a proof of the conjecture (ArXiv:1506.07056).

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Acknowledgements

Research of Jerrold R. Griggs was supported in part by a grant from the Simons Foundation (#282896 to Jerrold Griggs) and by a long-term visiting position at the IMA, University of Minnesota.

Wei-Tian Li was supported by Ministry of Science and Technology (No. 103-2115-M-005 -003 -MY2).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Jerrold R. Griggs

  2. Department of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan, 40227

    Wei-Tian Li

Authors
  1. Jerrold R. Griggs
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  2. Wei-Tian Li
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Correspondence to Jerrold R. Griggs .

Editor information

Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Griggs, J.R., Li, WT. (2016). Progress on poset-free families of subsets. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_14

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