Related papers: The Tur\'an Number of Surfaces

In this paper we estimate the planar Tur\'an number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when…

The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex…

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with…

A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, B\'ar\'any and F\"uredi asked to determine the maximum…

We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete…

We show that the Thurston seminorms of all finite covers of an aspherical 3-manifold determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic.

Given $\alpha>0$ and an integer $\ell\geq5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell^{-}$, a tight…

We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices…

Planar Tur\'an number, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. Ghosh, Gy\H{o}ri, Paulos and Xiao initiated the topic of the planar Tur\'an…

Given any $\varepsilon>0$ we prove that every sufficiently large $n$-vertex $3$-graph $H$ where every pair of vertices is contained in at least $(1/3+\varepsilon)n$ edges contains a copy of $C_{10}$, i.e.\ the tight cycle on $10$ vertices.…

The Tur\'an number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint…

Given a graph $H$ and a positive integer $n$, the {\it Tur\'an number} $\ex(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in(1,2)$ is called a {\it Tur\'an exponent} if…

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer $k\geq 3$, there exists a normal tiling of the Euclidean plane by convex…

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a…

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…

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Tur\'an number of Berge-$\mathcal{F}$ is the maximum…

An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists \alpha(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{\alpha(i)}$. The Tur\'an…

For fixed graphs $H$ and $F$, the \emph{generalized Tur\'an number} $\mathrm{ex}(n,H,F)$ is the maximum possible number of copies of a subgraph $H$ in an $n$-vertex $F$-free graph. This article is a survey of this extremal function whose…

Given positive integers $a\leq b \leq c$, let $K_{a,b,c}$ be the complete 3-partite 3-uniform hypergraph with three parts of sizes $a,b,c$. Let $H$ be a 3-uniform hypergraph on $n$ vertices where $n$ is divisible by $a+b+c$. We…

It follows from known results that every regular tripartite hypergraph of positive degree, with $n$ vertices in each class, has matching number at least $n/2$. This bound is best possible, and the extremal configuration is unique. Here we…