Abstract
A family \({\mathcal {F}}\) on ground set \([n]:=\{1,2,\ldots , n\}\) is maximal k-wise intersecting if every collection of at most k sets in \({\mathcal {F}}\) has non-empty intersection, and no other set can be added to \({\mathcal {F}}\) while maintaining this property. In 1974, Erdős and Kleitman asked for the minimum size of a maximal k-wise intersecting family. We answer their question for \(k=3\) and sufficiently large n. We show that the unique minimum family is obtained by partitioning the ground set [n] into two sets A and B with almost equal sizes and taking the family consisting of all the proper supersets of A and of B.
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References
Alon, N., Frankl, P.: The maximum number of disjoint pairs in a family of subsets. Graphs Combinatorics 1, 13–21 (1985)
Alon, N., Das, S., Glebov, R., Sudakov, B.: Comparable pairs in families of sets. J. Combinatorial Theory Series B 115, 164–185 (2015)
Balogh, J., Bushaw, N., Collares, M., Liu, H., Morris, R., Sharifzadeh, M.: The typical structure of graphs with no large cliques. Combinatorica 37, 617–632 (2017)
Balogh, J., Chen, C., Luo, H.: Maximal\(3\)-wise Intersecting Families with Minimum Size: the Odd Case, arXiv:2206.09334 (2022)
Boros, E., Füredi, Z., Kahn, J.: Maximal intersecting families and affine regular polygons in \(PG(2, q)\). J. Combinatorial Theory Series A 52(1), 1–9 (1989)
Bucić, M., Letzter, S., Sudakov, B., Tran, T.: Minimum saturated families of sets. Bull. Lond. Math. Soc. 50(4), 725–732 (2018)
Dow, S.J., Drake, D.A., Füredi, Z., Larson, J.A.: A lower bound for the cardinality of a maximal family of mutually intersecting sets of equal size. Proc. 16th Southeast. Int. Conf. Combinatorics Graph Theory Comput. 48, 47–48 (1985)
Dukes, P., Howard, L.: Small maximally disjoint union-free families. Discret. Math. 308(18), 4272–4275 (2008)
Ellis, D., Sudakov, B.: Generating all subsets of a finite set with disjoint unions. J. Combinatorial Theory Series A 118(8), 2319–2345 (2011)
Erdős, P.: On extremal problems of graphs and generalized graphs. Israel J. Math. 2(3), 183–190 (1964)
Erdős, P.: Problem sessions, I. Rival (Ed.), Ordered Sets, Proceedings of the NATO Advanced Study, Reidel, Dordrecht, 860–861 (1981)
Erdős, P., Lovász, L.: Problems and results on\(3\)-chromatic hypergraphs and some related questions, Colloquia Mathematica Societatis János Bolyai 10. Infinite and Finite Sets Keszthely (Hungary) (1973)
Erdős, P., Kleitman, D.J.: Extremal problems among subsets of a set. Discret. Math. 8, 281–294 (1974)
Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxford Second Series 12, 313–320 (1961)
Erdős, P., Hajnal, A., Moon, J.W.: A problem in graph theory. Am. Math. Mon. 71(1), 1107–1110 (1964)
Faudree, J., Faudree, R., Schmitt, J.R.: A survey of minimum saturated graphs. Electron. J. Combinatorics 18, 36 (2011)
Frankl, N., Kiselev, S., Kupavskii, A., Patkós, B.: VC-saturated set systems. Eur. J. Combinatorics 104, 10 (2022)
Gerbner, D., Keszegh, B., Lemons, N., Palmer, C., Pálvölgyi, D., Patkós, B.: Saturating Sperner families. Graphs Combinatorics 29(5), 1355–1364 (2013)
Guy, R.K.: A miscellany of Erdős problems. Am. Math. Mon. 90, 118–120 (1983)
Hendrey, K., Lund, B., Tompkins, C., Tran, T.: Maximal\(3\)-wise intersecting families, arXiv:2110.12708v1 (2021)
Janzer, B.: A note on saturation for\(k\)-wise intersecting families, Combinatorial Theory, 2(2), Paper No. 11, 5 pp (2022)
Kahn, J.: On a problem of Erdős and Lovász. J. Am. Math. Soc. 7(1), 125–143 (1994)
Acknowledgements
The authors are grateful for Jingwei Xu, Simon Piga and Andrew Treglown, who participated in fruitful discussions at the beginning of the project. Simon Piga and Andrew Treglown’s visit to University of Illinois was partially supported by NSF RTG grant DMS 1937241. We thank the anonymous referees for their careful reading of the manuscript and many useful comments.
Funding
József Balogh—Research is partially supported by NSF Grant DMS-1764123, NSF RTG grant DMS 1937241, Arnold O. Beckman Research Award (UIUC Campus Research Board RB 22000), and the Langan Scholar Fund (UIUC). Kevin Hendrey—This work was supported by the Institute for Basic Science (IBS-R029-C1). Ben Lund—This work was supported by the Institute for Basic Science (IBS-R029-C1). Haoran Luo—Research is partially supported by UIUC Campus Research Board RB 22000. Casey Tompkins—This work was supported by NKFIH grant K135800. Tuan Tran—This work was supported by the Institute for Basic Science (IBS-R029-Y1), and the Excellent Young Talents Program (Overseas) of the National Natural Science Foundation of China.
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Balogh, J., Chen, C., Hendrey, K. et al. Maximal 3-Wise Intersecting Families. Combinatorica 43, 1045–1066 (2023). https://doi.org/10.1007/s00493-023-00046-3
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DOI: https://doi.org/10.1007/s00493-023-00046-3