Related papers: Forbidden subgraphs in the norm graph

An interval graph is proper iff it has a representation in which no interval contains another. Fred Roberts characterized the proper interval graphs as those containing no induced star $K_{1,3}$. Proskurowski and Telle have studied…

We show that for each $k\geq 4$ and $n>r\geq k+1$, every $n$-vertex $r$-uniform hypergraph with no Berge cycle of length at least $k$ has at most $\frac{(k-1)(n-1)}{r}$ edges. The bound is exact, and we describe the extremal hypergraphs.…

Fix integers $r \ge 2$ and $1\le s_1\le \cdots \le s_{r-1}\le t$ and set $s=\prod_{i=1}^{r-1}s_i$. Let $K=K(s_1, \ldots, s_{r-1}, t)$ denote the complete $r$-partite $r$-uniform hypergraph with parts of size $s_1, \ldots, s_{r-1}, t$. We…

The theta graph $\Theta_{\ell,t}$ consists of two vertices joined by $t$ vertex-disjoint paths of length $\ell$ each. For fixed odd $\ell$ and large $t$, we show that the largest graph not containing $\Theta_{\ell,t}$ has at most $c_{\ell}…

We consider an infinite version of the bipartite Tur\'{a}n problem. Let $G$ be an infinite graph with $V(G) = \mathbb{N}$ and let $G_n$ be the $n$-vertex subgraph of $G$ induced by the vertices $\{1,2, \dots, n \}$. We show that if $G$ is…

For ordinary graphs it is known that any graph $G$ with more edges than the Tur{\'a}n number of $K_s$ must contain several copies of $K_s$, and a copy of $K_{s+1}^-$, the complete graph on $s+1$ vertices with one missing edge. Erd\H{o}s…

We study the typical structure of oriented graphs and digraphs that do not contain a blow-up T_{r+1}^t of a transitive tournament. For any integers r >= 2, t >= 1 and any real a in (3/2,2], we prove that almost all T_{r+1}^t-free oriented…

We prove that there exist graphs which do not contain $K_t$ as an odd minor and whose chromatic number is at least $(\frac 32-o(1))t$. This disproves, in a strong form, the odd Hadwiger conjecture of Gerards and Seymour from 1993.

In a recent paper, Hunter, Milojevi\'c, Sudakov and Tomon consider the maximum number of edges in an $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ and no induced copy of a "pattern" graph $H$. They conjecture…

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently…

Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$, and let $s(F):=\sup\{s: \exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\}$. Following recent work of…

In this short note, we prove the following analog of the K\H{o}v\'ari-S\'os-Tur\'an theorem for intersection graphs of boxes. If $G$ is the intersection graph of $n$ axis-parallel boxes in $\mathbb{R}^{d}$ such that $G$ contains no copy of…

Kalai conjectured that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r} {n \choose r-1}$ edges contains all tight $r$-trees of some fixed size $t$. We prove Kalai's conjecture for $r$-partite $r$-uniform hypergraphs.…

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite…

Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is called the generalized Tur\'{a}n number, denoted by $ex(n,T,F)$. When $T=K_2$, it reduces to the classical Tur\'{a}n number $ex(n,F)$.…

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper…

For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Tur\'{a}n number…

The generalized Kneser hypergraph $KG^{r}(n,k,s)$ is the hypergraph whose vertices are all the $k$-subsets of $\{1,\ldots ,n\}$, and edges are $r$-tuples of distinct vertices such that any pair of them has at most $s$ elements in their…

In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is the largest minimum degree $\delta(G)$ among all $r$-partite graphs $G$ with parts of size $n$ and…

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$.…