Related papers: Hypergraph based Berge hypergraphs

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…

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…

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,H,\mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex graphs that do not contain any member of $\mathcal{F}$ as a…

A Berge cycle of length $\ell$ in a hypergraph is an alternating sequence of $\ell$ distinct vertices and $\ell$ distinct edges $v_1,e_1,v_2, \ldots, v_\ell, e_{\ell}$ such that $\{v_i, v_{i+1}\} \subseteq e_i$ for all $i$, with indices…

In this paper we introduce a unifying approach to the generalized Tur\'an problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an…

Let $\mathcal{H}$ be a 3-graph on $n$ vertices. The matching number $\nu(\mathcal{H})$ is defined as the maximum number of disjoint edges in $\mathcal{H}$. The generalized triangle $F_5$ is a 3-graph on the vertex set $\{a,b,c,d,e\}$ with…

Let $\mathcal{H}$ be a set of given connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no $H$ as an induced subgraph for any $H\in \mathcal{H}$. The graph $G$ is super-edge-connected if each minimum edge-cut…

An $r$-uniform hypergraph ($r$-graph) is linear if any two edges intersect at most one vertex. For a graph $F$, a hypergraph $H$ is Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for all $e$ in…

The forbidden subgraph problem is among the oldest in extremal combinatorics -- how many edges can an $n$-vertex $F$-free graph have? The answer to this question is the well-studied extremal number of $F$. Observing that every extremal…

We say that a hypergraph $\mathcal{H}$ contains a graph $H$ as a trace if there exists some set $S\subset V(\mathcal{H})$ such that $\mathcal{H}|_S=\{h\cap S: h\in E(\mathcal{H})\}$ contains a subhypergraph isomorphic to $H$. We study the…

A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i},v_{i+1} \in e_i$, for $i \le k$. F\"uredi, Kostochka and Luo, and independently…

A classical conjecture of Erd\H{o}s and S\'os asks to determine the Tur\'an number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, for all $k$ and $r$, with $r \ge k…

The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…

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…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of…

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq \phi(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the…

For a fixed graph $F$, let $ex_F(G)$ denote the size of the largest $F$-free subgraph of $G$. Computing or estimating $ex_F(G)$ for various pairs $F,G$ is one of the central problems in extremal combinatorics. It is thus natural to ask how…

A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least 5 or the complement of one.…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

Let $F$ be a strictly $k$-balanced $k$-uniform hypergraph with $e(F)\geq |F|-k+1$ and maximum co-degree at least two. The random greedy $F$-free process constructs a maximal $F$-free hypergraph as follows. Consider a random ordering of the…