Related papers: Erd\H{o}s-Ko-Rado for random hypergraphs: asymptot…

Let G=PGL(2,q) be the projective general linear group acting on the projective line P_q. A subset S of G is intersecting if for any pair of permutations \pi,\sigma in S, there is a projective point p in P_q such that p^\pi=p^\sigma. We…

In the random geometric graph model $\mathsf{Geo}_d(n,p)$, we identify each of our $n$ vertices with an independently and uniformly sampled vector from the $d$-dimensional unit sphere, and we connect pairs of vertices whose vectors are…

For an integer $r \ge 3$ and a subset $L \subset [0,r-1]$, a graph $G$ is $(K_{r}, L)$-intersecting if the number of vertices in the intersection of every pair of $K_r$ in $G$ belongs to $L$. We study the maximum number of $K_r$ in an…

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next,…

We consider the near-critical Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and $A$ is large,…

We show that if a simplicial complex is a near-cone of sufficiently high depth, then the only maximum families of small pairwise intersecting faces are those with a common intersection. Thus, near-cones of sufficiently high depth satisfy…

We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called…

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, itself an extension of the $t$-intersecting Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum size of a…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether…

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the…

We consider the problem of detecting a community of densely connected vertices in a high-dimensional bipartite graph of size $n_1 \times n_2$. Under the null hypothesis, the observed graph is drawn from a bipartite Erd\H{o}s-Renyi…

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50--62] we used properties of empirical distributions of independent random variables…

A family of axis-aligned boxes in $\er^d$ is \emph{$k$-neighborly} if the intersection of every two of them has dimension at least $d-k$ and at most $d-1$. Let $n(k,d)$ denote the maximum size of such a family. It is known that $n(k,d)$ can…

We study the weak $K_s$-saturation number of the Erd\H{o}s--R\'{e}nyi random graph $\mathbbmsl{G}(n, p)$, denoted by $\mathrm{wsat}(\mathbbmsl{G}(n, p), K_s)$, where $K_s$ is the complete graph on $s$ vertices. Kor\'{a}ndi and Sudakov in…

For graphs $G$ and $H$, let $G {\displaystyle\smash{\begin{subarray}{c} \hbox{$\tiny\rm rb$} \\ \longrightarrow \\ \hbox{$\tiny\rm p$} \end{subarray}}}H$ denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$…

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with…

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic…