Related papers: 2-cancellative hypergraphs and codes

A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq…

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…

For a family $\mathcal F$ define $\nu(\mathcal F,t)$ as the largest $s$ for which there exist $A_1,\ldots, A_{s}\in \mathcal F$ such that for $i\ne j$ we have $|A_i\cap A_j|< t$. What is the largest family $\mathcal F\subset{[n]\choose k}$…

We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem…

We make some progress on a question of Babai from the 1970s, namely: for $n, k \in \mathbb{N}$ with $k \le n/2$, what is the largest possible cardinality $s(n,k)$ of an intersecting family of $k$-element subsets of $\{1,2,\ldots,n\}$…

We say that a set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set…

The generalized Kneser graph $K(n,k,t)$ for integers $k>t>0$ and $n>2k-t$ is the graph whose vertices are the $k$-subsets of $\{1,\dots,n\}$ with two vertices adjacent if and only if they share less than $t$ elements. We determine the…

We show that the biggest possible average set size in the complement $2^{\{1,2,\ldots, n\}} \setminus A$ of a union-closed family $A \subset 2^{\{1,2, \ldots, n\}}$ is $\tfrac{n+1}{2}$. With the same proof we get a sharp upper bound for the…

A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in…

A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most…

Let $3\le d\le k$ and $\nu\ge 0$ be fixed and $\mathcal{F}\subset\binom{[n]}{k}$. The matching number of $\mathcal{F}$, denoted by $\nu(\mathcal{F})$, is the maximum number of pairwise disjoint sets in $\mathcal{F}$, and $\mathcal{F}$ is…

We define an all-$k$-isolating set of a graph to be a set $S$ of vertices such that, if one removes $S$ and all its neighbors, then no component in what remains has order $k$ or more. The case $k=1$ corresponds to a dominating set and the…

Given a graph $H$, we say that a graph $G$ is $H$-saturated if $G$ contains no copy of $H$ but adding any new edge to $G$ creates a copy of $H$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices…

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise…

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…

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no…

Two families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$ are called cross-$t$-intersecting if $|A\cap B|\geq t$ for all $A\in\mathcal A$, $B\in\mathcal B$. Let $n$, $k$ and $\ell$ be positive integers such…