22 research outputs found

    Minimum saturated families of sets

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    We call a family F\mathcal{F} of subsets of [n][n] ss-saturated if it contains no ss pairwise disjoint sets, and moreover no set can be added to F\mathcal{F} while preserving this property (here [n]={1,,n}[n] = \{1,\ldots,n\}). More than 40 years ago, Erd\H{o}s and Kleitman conjectured that an ss-saturated family of subsets of [n][n] has size at least (12(s1))2n(1 - 2^{-(s-1)})2^n. It is easy to show that every ss-saturated family has size at least 122n\frac{1}{2}\cdot 2^n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2+ε)2n(1/2 + \varepsilon)2^n, for some fixed ε>0\varepsilon > 0, seems difficult. In this note, we prove such a result, showing that every ss-saturated family of subsets of [n][n] has size at least (11/s)2n(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 F1++Fs|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| where F1,,Fs\mathcal{F}_1, \ldots, \mathcal{F}_s are families of subsets of [n][n], such that there are no ss pairwise disjoint sets, one from each family Fi\mathcal{F}_i, and furthermore no set can be added to any of the families while preserving this property. We show that F1++Fs(s1)2n|\mathcal{F}_1| + \ldots + |\mathcal{F}_s| \ge (s-1)\cdot 2^n, which is tight e.g.\ by taking F1\mathcal{F}_1 to be empty, and letting the remaining families be the families of all subsets of [n][n].Comment: 8 page

    Bounds on three- and higher-distance sets

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    A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for fixed s and M. In this paper, we improve the known upper bound for s-distance sets in n-sphere for s=3,4. In particular, we determine the maximum cardinalities of three-distance sets for n=7 and 21. We also give the maximum cardinalities of s-distance sets in the Hamming space and the Johnson space for several s and dimensions.Comment: 12 page

    Multiply intersecting families of sets

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    AbstractLet [n] denote the set {1,2,…,n}, 2[n] the collection of all subsets of [n] and F⊂2[n] be a family. The maximum of |F| is studied if any r subsets have an at least s-element intersection and there are no ℓ subsets containing t+1 common elements. We show that |F|⩽∑i=0t−sn−si+t+ℓ−st+2−sn−st+1−s+ℓ−2 and this bound is asymptotically the best possible as n→∞ and t⩾2s⩾2, r,ℓ⩾2 are fixed

    On LL-close Sperner systems

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    For a set LL of positive integers, a set system F2[n]\mathcal{F} \subseteq 2^{[n]} is said to be LL-close Sperner, if for any pair F,GF,G of distinct sets in F\mathcal{F} the skew distance sd(F,G)=min{FG,GF}sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\} belongs to LL. We reprove an extremal result of Boros, Gurvich, and Milani\v c on the maximum size of LL-close Sperner set systems for L={1}L=\{1\} and generalize to L=1|L|=1 and obtain slightly weaker bounds for arbitrary LL. We also consider the problem when LL might include 0 and reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set systems with all skew distances belonging to L={0,1}L=\{0,1\}
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