Related papers: Regular Tur\'an numbers and some Gan-Loh-Sudakov-t…

We investigate natural Tur\'an problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural \textit{Tur\'an density coefficient} that measures how large a fraction of directed…

The oriented Tur\'{a}n number of a given oriented graph $\overrightarrow{F}$, denoted by $\exo(n,\overrightarrow{F})$, is the largest number of arcs in $n$-vertex $\overrightarrow{F}$-free oriented graphs. This parameter could be seen as a…

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $r-2$ new vertices. Keevash and Sidorenko~ have…

The classical K\H{o}v\'ari-S\'os-Tur\'an theorem states that if $G$ is an $n$-vertex graph with no copy of $K_{s,t}$ as a subgraph, then the number of edges in $G$ is at most $O(n^{2-1/s})$. We prove that if one forbids $K_{s,t}$ as an…

We propose a strengthening of the conclusion in Tur\'an's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Tur\'an theorem by weakening its…

We introduce a modification of the Tur\'an density of ordered graphs and investigate this graph parameter.

The regular Tur\'an number of a graph $F$, denoted by rex$(n,F)$, is the largest number of edges in a regular graph $G$ of order $n$ such that $G$ does not contain subgraphs isomorphic to $F$. Giving a partial answer to a recent problem…

We prove Tur\'an-type theorems for two related Ramsey problems raised by Bollob\'as and by Fox and Sudakov. First, for $t \ge 3$, we show that any two-colouring of the complete graph on $n$ vertices that is $\delta$-far from being…

In this paper we initiate a systematic study of the Tur\'an problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the…

As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…

We study the Tur\'an number of long cycles in random graphs and in pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the…

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,…

We study the following generalization of the Tur\'an problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of…

Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove…

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains…

In a generalized Tur\'an problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Tur\'an problems where the host graph is planar. In particular we…

The oriented Tur\'{a}n number of a given oriented graph $\overrightarrow{F}$, denoted by $\exo(n,\overrightarrow{F})$, is the largest number of arcs in $n$-vertex $\overrightarrow{F}$-free oriented graphs. This concept could be seen as an…

Given $r$-uniform hypergraphs $G$ and $H$ the Tur\'an number $\rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the Erd\H{o}s-R\'enyi random…

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…

Combinatorial batch codes provide a tool for distributed data storage, with the feature of keeping privacy during information retrieval. Recently, Balachandran and Bhattacharya observed that the problem of constructing such uniform codes in…