Related papers: Counting Hypergraphs with Large Girth

Let $\mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m \times n$ incidence matrix of $\mathcal{H}$ and let us denote $\lambda =\max_{v \perp \overline{1},\|v\| = 1}\|Mv\|$. We show that the…

Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that $G$ contains a large subgraph with high…

Erd\H{o}s proved an upper bound on the number of edges in an $n$-vertex non-Hamiltonian graph with given minimum degree and showed sharpness via two members of a particular graph family. F\"{u}redi, Kostochka and Luo showed that these two…

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph $T$ consisting of three edges $e,f$ and $g$ such that $|e \cap f| = |f…

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…

We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair…

Recently, Mubayi and Wang showed that for $r\ge 4$ and $\ell \ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $\ell$ is at most $2^{O( n^{r-1} (\log n)^{(r-3)/(r-2)})}$. We improve this bound to…

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…

The `global' Zarankiewicz problem for hypergraphs asks for an upper bound on the number of edges of a finite $r$-hypergraph $V$ in terms of the number $|V|$ of its vertices, assuming the edge relation is induced by a fixed $K_{k, \dots,…

One of the most basic questions one can ask about a graph $H$ is: how many $H$-free graphs on $n$ vertices are there? For non-bipartite $H$, the answer to this question has been well-understood since 1986, when Erd\H{o}s, Frankl and R\"odl…

The main purpose of the paper is to construct a sequence of graphs of constant degree with indefinitely growing girths admitting embeddings into $\ell_1$ with uniformly bounded distortions. This result answers the problem posed by N.…

Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $\chi(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of…

For an $r$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size of a set~$M$ of edges in $H$ such that every two edges in $M$ intersect in less than $m$ vertices, and let $\tau^{(m)}(H)$ denote the minimum size of a collection…

A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine…

We study the question of the least number of random edges that need to be added to a P\'osa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle,…

Frankl and F\"uredi (1989) conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest graph-Lagrangian of all $r$-graphs with $m$ edges. In this paper, we…

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

We give an upper bound for the maximum number of edges in an $n$-vertex 2-connected $r$-uniform hypergraph with no Berge cycle of length $k$ or greater, where $n\geq k \geq 4r\geq 12$. For $n$ large with respect to $r$ and $k$, this bound…