Bucić, M;
Letzter, S;
Sudakov, B;
Tran, T;
(2018)
Minimum saturated families of sets.
Bulletin of the London Mathematical Society
, 50
(4)
pp. 725-732.
10.1112/blms.12184.
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Abstract
We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}). More than 40 years ago, Erd˝os and Kleitman conjectured that an s-saturated family of subsets of [n] has size at least (1 − 2 −(s−1))2n. It is easy to show that every s-saturated family has size at least 1 2 · 2 n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2 + ε)2n, for some fixed ε > 0, seems difficult. In this note, we prove such a result, showing that every s-saturated family of subsets of [n] has size at least (1 − 1/s)2n. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on |F1| + . . . + |Fs| where F1, . . . , Fs are families of subsets of [n], such that there are no s pairwise disjoint sets, one from each family Fi , and furthermore no set can be added to any of the families while preserving this property. We show that |F1| + . . . + |Fs| ≥ (s − 1) · 2 n, which is tight e.g. by taking F1 to be empty, and letting the remaining families be the families of all subsets of [n].
Type: | Article |
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Title: | Minimum saturated families of sets |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.1112/blms.12184 |
Publisher version: | http://dx.doi.org/10.1112/blms.12184 |
Language: | English |
Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
URI: | https://discovery.ucl.ac.uk/id/eprint/10107224 |
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