Related papers: The number of hypergraphs without linear cycles

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…

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, then the complete $k$-uniform hypergraph on $n$ vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an…

In 1996, in his last paper, Erd\H{o}s asked the following question that he formulated together with Faudree: is there a positive $c$ such that any $(n+1)$-regular graph $G$ on $2n$ vertices contains at least $c 2^{2n}$ distinct…

How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erd\H{o}s about $27$ years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still…

The size-Ramsey number $R^{(k)}(H)$ of a $k$-uniform hypergraph $H$ is the minimum number of edges in a $k$-uniform hypergraph $G$ with the property that every `$2$-edge coloring' of $G$ contains a monochromatic copy of $H$. For $k\ge2$ and…

We prove a conjecture of K\"uhn, Osthus, Townsend and Zhao \cite{kuhn2017structure} stating that almost every $C_k$-free oriented graph on $n$ vertices has $\Theta(n)$ backwards edges in a transitive-optimal ordering. The same holds for…

Given a graph $F$, a hypergraph is called a Berge-$F$ if it can be obtained by expanding each edge of $F$ into a hyperedge containing it. Let $M_{k}$ denote the matching of size $k$. Kang, Ni, and Shan [12] determined the Tur\'an number of…

Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is…

Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n,K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It…

Let $k\geq 2$ and fix a $k$-uniform hypergraph $\mathcal{F}$. Consider the random process that, starting from a $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices, repeatedly deletes the edges of a copy of $\mathcal{F}$ chosen uniformly…

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of…

Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} +…

In this paper we consider the problem to reconstruct a $k$-uniform hypergraph from its line graph. In general this problem is hard. We solve this problem when the number of hyperedges containing any pair of vertices is bounded. Given an…

The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete…

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform…

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…

Recently Lin, Wang and Zhou have proved that every $3$-connected nonbipartite graph of minimum degree at least $k$ with $k\ge 6$ and order at least $k+2$ contains $k$ cycles of consecutive lengths. They also conjecture that this result is…

We consider edge decompositions of $K_v^{(3)}-I$, the complete 3-uniform hypergraph of order $v$ minus a 1-factor (parallel class, packing of $v/3$ disjoint edges). We prove that a decomposition into tight 6-cycles exists if and only if…

It has been conjectured that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r - 1)-coloring of the edges of the complete r-uniform hypergraph on n vertices. In this paper, we show that…

It is conjectured by Frankl and F\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$…