Related papers: Sidon sets and $C_4$-saturated graphs

The saturation number $\text{sat}_r(n,\mathcal{F})$ is the minimum number of hyperedges in an $r$-uniform $\mathcal{F}$-saturated hypergraph on $n$ vertices. We determine this parameter for $3$-uniform Berge-$K_4$ hypergraphs, proving that…

Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the…

For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs…

For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$…

Let $G$ be a graph with degree sequence $d_1,d_2,\ldots,d_n$. Given a positive integer $p$, denote by $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given an integer $p$, how large can $e_p(G)$…

Let $\mathcal{F}$ be a family of graphs. The Tur\'{a}n number $ex(n;\mathcal{F})$ is defined to be the maximum number of edges in a graph of order $n$ that is $\mathcal{F}$-free. In 1959, Erd\H{o}s and Gallai determined the Tur\'an number…

The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the…

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is called \textit{$\mathcal{F}$-free} if for any $F\in \mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized…

Let $C_{\ell}$ be the cycle of order ${\ell}$. The square of $C_{\ell}$, denoted by $C_{\ell}^2$, is obtained by joining all pairs of vertices with distance no more than two in $C_{\ell}$. A graph is called $F$-free if it does not contain…

Given a graph $F$, the planar Tur\'an number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining…

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…

We consider a natural generalisation of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem by studying the maximum number $ex_F(n,G)$ of edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground set…

A vertex set $D$ in a finite undirected graph $G$ is an efficient dominating set (e.d.s. for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence…

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…

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$ vertices…

In 1984, Erd\H{o}s and Simonovits conjectured the following: given a bipartite graph $H$, there exist constants $\beta, C > 0$ such that any graph $G$ on $n$ vertices and $pn^2\geq C \mathrm{ex}(n, H)$ edges contains at least $\beta…

Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when…

Let $\varphi(x_1,\ldots, x_h) = c_1 x_1 + \cdots + c_h x_h $ be a linear form with coefficients in a field $\mathbf{F}$, and let $V$ be a vector space over $\mathbf{F}$. A nonempty subset $A$ of $V$ is a $\varphi$-Sidon set if, for all…

For graphs $G$ and $H$, we say that $G$ is $H$-free if no induced subgraph of $G$ is isomorphic to $H$, and that $G$ is $H$-induced-saturated if $G$ is $H$-free but removing or adding any edge in $G$ creates an induced copy of $H$. A full…

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In…