Related papers: On the Chromatic Thresholds of Hypergraphs

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…

In the 1980s, Erd\H{o}s and S\'os first introduced an extremal problem on hypergraphs with density constraints. Given an $r$-uniform hypergraph $F$ (or $r$-graph for short), its uniform Tur\'an density $\pi_u(F)$ is the smallest value of…

Given a graph $H,$ we say that a graph is \textit{$H$-free} if it 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 set of all the corresponding…

In 1973, Erd\H{o}s and Simonovits asked whether every $n$-vertex triangle-free graph with minimum degree greater than $1/3 \cdot n$ is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for…

Fix a color-critical graph $H$ with $\chi(H)=r+1\geq 3$. Simonovits' chromatic critical edge theorem and Nikiforov's spectral chromatic critical edge theorem imply that $T_{n,r}$ is the extremal graph with the maximum size and the maximum…

We construct a hereditary class of triangle-free graphs with unbounded chromatic number, in which every non-trivial graph either contains a pair of non-adjacent twins or has an edgeless vertex cutset of size at most two. This answers in the…

In this paper, we initiate a systematic study on a new notion called near optimal colourability which is closely related to perfect graphs and the Lov{\'a}sz theta function. A graph family $\mathcal{G}$ is {\em near optimal colourable} if…

An $r$-hued coloring of a simple graph $G$ is a proper coloring of its vertices such that every vertex $v$ is adjacent to at least $\min\{r, \deg(v)\}$ differently colored vertices. The minimum number of colors needed for an $r$-hued…

Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, determine the infimum of $\alpha$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at…

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universal if $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced) subgraph. For the family of all graphs on $n$ vertices and of maximum degree at most…

Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey…

Given a hypergraph $F$, what is the largest chromatic number that an $F$-free hypergraph can have? In the case of graphs, this question is easy to answer: the chromatic number is unbounded if $F$ contains a cycle, and the largest chromatic…

In 2011, Meunier conjectured that for positive integers $n,k,r,s$ with $ k\geq 2$, $r\geq 2$, and $n\geq \max (\{r,s\})k$, the chromatic number of $s$ -stable $r$-uniform Kneser hypergraphs is equal to $\left\lceil \frac{n-\max…

The uniform Tur\'an density $\pi_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining…

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study…

In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the…

Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist injections $\psi:U\to V$ and $\varphi:E\to \mathcal{E}$ such that for every $e=\{u,v\}\in E$,…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H.…

Given a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a {\em Berge-$F$} if there is a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform…