Related papers: Vanishing codegree Tur\'{a}n density implies vanis…

In this paper we describe a number of extensions to Razborov's semidefinite flag algebra method. We will begin by showing how to apply the method to significantly improve the upper bounds of edge and vertex Tur\'an density type results for…

A result of Gy\'arf\'as says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all…

We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $\delta_1(G)>d$ then every…

Given graphs $H$ and $F$, $\mathrm{ex}(n,H,F)$ denotes the largest number of copies of $H$ in $F$-free $n$-vertex graphs. Let $\chi(H)<\chi(F)=r+1$. We say that $H$ is $F$-Tur\'an-stable if the following holds. For any $\varepsilon>0$ there…

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this…

A density $\alpha\in [0, 1)$ is a jump for $r$ if there is some $c >0$ such that there does not exist a family of $r$-uniform hypergraphs $\mathcal{F}$ with Tur\'an density $\pi(\mathcal{F})$ in $(\alpha, \alpha + c)$. Erd\"os conjectured…

A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\cal{F}}$ of hypergraphs, the linear Tur\'an number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform…

We show that there is a constant $c$ such that any 3-uniform hypergraph $\mathcal H$ with $n$ vertices and at least $cn^{5/2}$ edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of…

Given integers $r > t \ge 1$ and a real number $p > 0$, the $(t,p)$-norm $\left\lVert \mathcal{H} \right\rVert_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th power of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T…

The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph…

The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key…

This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this…

For a graph, $G$, and a vertex $v \in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \in V(G)$, the vertex sets…

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted $\alpha_r(G)$ is the largest size of a…

Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs…

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and…

In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many…

Given positive integers k and r where 4 divides k and k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who…

In this paper we introduce a unifying approach to the generalized Tur\'an problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an…

Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| \geq (\frac{k}{2} - \frac{1}{k-1})|V(G)| - \frac{k(k-3)}{2(k-1)}$. The…