Related papers: Some extremal results on complete degenerate hyper…

Let $\mathcal{H}$ be a family of graphs. The generalized Tur\'an number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of…

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine…

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if…

The generalized Tur\'{a}n number $\mathrm{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. For an integer $t \geq 1$, let $tF$ be the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku,…

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge…

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be…

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\'{a}n graph on $n$ vertices with a maximum clique of order…

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…

We introduce the following simpler variant of the Tur\'an problem: Given integers $n>k>r\geq 2$ and $m\geq 1$, what is the smallest integer $t$ for which there exists an $r$-uniform hypergraph with $n$ vertices, $t$ edges and $m$ connected…

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Tur\'an number $\mathrm{ex}(n,F)$ that is among the most studied…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The general Tur\'an number, denoted by $ex(n, H,\mathscr{F})$, is the maximum number of copies of $H$…

For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently large…

Given an $r_0$-uniform hypergraph $F$, we define its $r$-uniform expansion $F^{(r)}$ to be the hypergraph obtained from $F$ by inserting $r-r_0$ distinct vertices into each edge of $F$, and we define $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ to be…

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free…

Tur\'an's famous tetrahedron problem is to compute the Tur\'an density of the tetrahedron $K_4^3$. This is equivalent to determining the maximum $\ell_1$-norm of the codegree vector of a $K_4^3$-free $n$-vertex $3$-uniform hypergraph. We…

In this note, we prove several Tur\'an-type results on geometric hypergraphs. The two main theorems are 1) Every $n$-vertex geometric 3-hypergraph in 2-space with no three strongly crossing edges has at most $O(n^2)$ edges, 2) Every…

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure…

Given a tree T on v vertices and an integer k exceeding one. One can define the k-expansion T^k as a k-uniform linear hypergraph by enlarging each edge with a new, distinct set of (k-2) vertices. Then T^k has v+ (v-1)(k-2) vertices. The aim…

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…

Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$…