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- W2783809489 abstract "We call a family $mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $mathcal{F}$ while preserving this property (here $[n] = {1,ldots,n}$). More than 40 years ago, ErdH{o}s and Kleitman conjectured that an $s$-saturated family of subsets of $[n]$ has size at least $(1 - 2^{-(s-1)})2^n$. It is easy to show that every $s$-saturated family has size at least $frac{1}{2}cdot 2^n$, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of $(1/2 + varepsilon)2^n$, for some fixed $varepsilon > 0$, seems difficult. In this note, we prove such a result, showing that every $s$-saturated family of subsets of $[n]$ has size at least $(1 - 1/s)2^n$. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on $|mathcal{F}_1| + ldots + |mathcal{F}_s|$ where $mathcal{F}_1, ldots, mathcal{F}_s$ are families of subsets of $[n]$, such that there are no $s$ pairwise disjoint sets, one from each family $mathcal{F}_i$, and furthermore no set can be added to any of the families while preserving this property. We show that $|mathcal{F}_1| + ldots + |mathcal{F}_s| ge (s-1)cdot 2^n$, which is tight e.g. by taking $mathcal{F}_1$ to be empty, and letting the remaining families be the families of all subsets of $[n]$." @default.
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- W2783809489 date "2018-07-15" @default.
- W2783809489 modified "2023-09-25" @default.
- W2783809489 title "Minimum saturated families of sets" @default.
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- W2783809489 doi "https://doi.org/10.1112/blms.12184" @default.
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