Related papers: On Saturated $k$-Sperner Systems

For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any…

If X is an n-element set, we call a family G of subsets of X a k-generator for X if every subset of X can be expressed as a union of at most k disjoint sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest…

A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$…

To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a…

For a given number of $k$-sets, how should we choose them so as to minimize the union-closed family that they generate? Our main aim in this paper is to show that, if $\mathcal{A}$ is a family of $k$-sets of size $\binom{t}{k}$, and $t$ is…

Extending a classical theorem of Sperner, we characterize the integers $m$ such that there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion. As an…

For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\…

A family ${\mathcal A}$ of $k$-subsets of $\{1,2,\dots, N\}$ is a Sidon system if the sumsets $A+B$, $A,B\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$…

Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a poset whose Hasse diagram consists of $s$ pairwise incomparable vertices in the upper layer and $t$ pairwise incomparable vertices in the lower layer, such that…

Let $p$ be a prime number. A saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is said to be supersolvable if there is a series $1 = S_0 \le S_1 \le \dots \le S_m = S$ of subgroups of $S$ such that $S_i$ is strongly…

A family of sets is called $r$-\emph{cover free} if no set in the family is contained in the union of $r$ (or less) other sets in the family. A $1$-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner's…

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…

There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptographic and communications. This work studies the generalization of cover-free families from sets to…

An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that…

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=\lambda \mathbf{x}…

In this paper we deal with the problem of finding the smallest and the largest elements of a totally ordered set of size $n$ using pairwise comparisons if $k$ of the comparisons might be erroneous where $k$ is a fixed constant. We prove…

Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$, but the addition of any edge $e\notin E(G)$ creates at least one copy…

Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal…

We prove, when $S$ is a $2$-group of order at most $2^9$, that each reduced fusion system over $S$ is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a $2$-group of order at…