Minimum saturated families of sets

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, Erd\H{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 $[n]$.Comment: 8 page

Bounds on three- and higher-distance sets

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

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

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