Related papers: Maximum size intersecting families of bounded mini…

For a hypergraph $\mathcal{H}$, define the minimum positive codegree $\delta_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$…

The \textit{minimum positive co-degree} of a non-empty $r$-graph $H$, denoted $\delta_{r-1}^+(H)$, is the largest integer $k$ such that if a set $S \subset V(H)$ of size $r-1$ is contained in at least one $r$-edge of $H$, then $S$ is…

The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by $\delta_{r-1}^+(H)$, is the largest integer $k$ such that for every $(r-1)$-set $S \subset V(H)$, if $S$ is contained in a hyperedge of $H$, then $S$ is contained in at…

We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to…

The \emph{minimum positive co-degree} of a non-empty $r$-graph ${H}$, denoted $\delta_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $ {H}$, then $S$ is contained in at least $k$ distinct…

We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges.…

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $\beta>0$, there exists an integer…

Let $H$ be a $k$-uniform hypergraph on $n$ vertices where $n$ is a sufficiently large integer not divisible by $k$. We prove that if the minimum $(k-1)$-degree of $H$ is at least $\lfloor n/k \rfloor$, then $H$ contains a matching with…

Let $M_k$ be a $2n$-vertex graph with $n$ pairwise disjoint edges and let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that span exactly $p$ edges and $s$ isolated vertices. We prove that for $n\ge 2p+s$ this family has the…

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$,…

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

The celebrated {Erd\H{o}s-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for…

The celebrated Erd\H{o}s-Ko-Rado theorem determines the maximum size of a $k$-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a $k$-uniform intersecting family that is not a subfamily of the so-called…

The celebrated Erd\H{o}s-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona,…

Let $\mathcal{H} \subseteq \binom{[n]}{r}$ be an $r$-uniform hypergraph on vertex set $[n] = \{1,2,\dots, n\}$. For an $r$-set of vertices $S \subseteq [n]$, the \emph{degree} of $S$ is defined as $\textrm{deg}(S)=\sum_{v \in…

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most…

We consider an Erd\H{o}s-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with…

The celebrated Erd\H{o}s-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform…

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erd\H{o}s-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding…