Related papers: New Tur\'an exponents for two extremal hypergraph …

Let $\mathrm{ex}(n, H)$ be the Tur\'{a}n number of $H$ for a given graph $H$. A graph is color-critical if it contains an edge whose removal reduces its chromatic number. Simonovits' chromatic critical edge theorem states that if $H$ is…

Given a graph $L$, the Tur\'an number $\textrm{ex}(n,L)$ is the maximum possible number of edges in an $n$-vertex $L$-free graph. The study of Tur\'an number of graphs is a central topic in extremal graph theory. Although the celebrated…

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results…

We address a problem which is a generalization of Tur\'an-type problems recently introduced by Imolay, Karl, Nagy and V\'ali. Let $F$ be a fixed graph and let $G$ be the union of $k$ edge-disjoint copies of $F$, namely $G =…

Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an number of graphs is a central…

Let $C_{\ell}$ be the cycle of order ${\ell}$. The square of $C_{\ell}$, denoted by $C_{\ell}^2$, is obtained by joining all pairs of vertices with distance no more than two in $C_{\ell}$. A graph is called $F$-free if it does not contain…

An $r$-graph is a triangle if there exists a positive integer $i \le \lceil r/2 \rceil$ such that it is isomorphic to the following $r$-graph with three edges: \begin{align*} \left\{\{1, \ldots, r\},~\{1, \ldots, i, r+1, \ldots,…

In this paper we present a novel approach in extremal set theory which may be viewed as an asymmetric version of Katona's permutation method. We use it to find more Tur\'an numbers of hypergraphs in the Erd\H{o}s--Ko--Rado range. An…

Let $r\ge 3$. Given an $r$-graph $H$, the minimum codegree $\delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Tur\'an density…

Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…

Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called…

Given two graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a…

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…

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a…

In 1975, Erd\H{o}s asked for the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Tur\'an-type results can be used to prove an upper bound of…

An $r$-uniform hypergraph has $(q,p)$-property if any set of $q$ vertices spans a complete sub-hypergraph on $p$ vertices. Let $t_r(n,q,p)$ be the minimum edge density of an $n$-vertex $r$-uniform hypergraph with {\em $(q,p)$-property} and…

The $r$-uniform linear $k$-cycle $C^r_k$ is the $r$-uniform hypergraph on $k(r-1)$ vertices whose edges are sets of $r$ consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges…

Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The…

Let $k \ge 2$ be an integer. We show that if $s = 2$ and $t \ge 2$, or $s = t = 3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s +…

The classical extremal function for a graph $H$, $ex(K_n, H)$ is the largest number of edges in a subgraph of $K_n$ that contains no subgraph isomorphic to $H$. Note that defining $ex(K_n, H-ind)$ by forbidding induced subgraphs isomorphic…