Related papers: Asymptotics of the hypergraph bipartite Tur\'an pr…

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 Tur\'an number of a graph $F$, $ex(n,F)$, is the maximum number of edges in a graph on $n$ vertices which does not contain $F$ as a subgraph. Let $S_{a,b}$ denote a double star with a central edge $uv$, $a$ leaves connected to $u$ and…

Given a hypergraph $\mathcal{H}$ and a graph $G$, we say that $\mathcal{H}$ is a \textit{Berge}-$G$ if there is a bijection between the hyperedges of $\mathcal{H}$ and the edges of $G$ such that each hyperedge contains its image. We denote…

Let $n,s,$ and $k$ be positive integers such that $k\geq 3$, $s\geq 3$ and $n\geq ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $\textrm{ar}(n,k,M_s)$ of an…

A long-standing conjecture of Erd\H{o}s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=\Theta(n^r)$. So far this conjecture is known to be true only for rationals of…

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…

In this note we describe how Laso\'n's generalization of Alon's Combinatorial Nullstellensatz gives a framework for constructing lower bounds on the Tur\'an number $\operatorname{ex}(n, K^{(r)}_{s_1,\dots,s_r})$ of the complete $r$-partite…

Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of…

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 $H$…

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set…

We study the generalized Tur\'an function $ex(n,H,F)$, when $H$ or $F$ is $K_{2,t}$. We determine the order of magnitude of $ex(n,H,K_{2,t})$ when $H$ is a tree, and determine its asymptotics for a large class of trees. We also determine…

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph…

For a given graph $F$, the $r$-uniform suspension of $F$ is the $r$-uniform hypergraph obtained from $F$ by taking $r-2$ new vertices and adding them to every edge. In this paper, we consider Tur\'{a}n problems on suspension hypergraphs,…

Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let…

The present paper is concerned with the various algebraic structures supported by the set of Tur\'an densities. We prove that the set of Tur\'an densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a…

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 Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine…

Classical questions in extremal graph theory concern the asymptotics of $\operatorname{ex}(G, \mathcal{H})$ where $\mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard' increasing sequence of host graphs $(G_1,…

In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities…

Given graphs $T$ and $H$, the generalized Tur\'an number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of…