Related papers: Supersaturated sparse graphs and hypergraphs

We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of Hajek [IEEE…

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n…

We investigate the following generalisation of the 'multiplication table problem' of Erd\H{o}s: given a bipartite graph with $m$ edges, how large is the set of sizes of its induced subgraphs? Erd\H{o}s's problem of estimating the number of…

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical…

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In…

In 2003, Fischermann et al. considered the maximum size of \textit{uniquely-dominatable} graphs, graphs whose dominating number is realized only by a unique dominating set. They conjectured a size bound and provide a general graph…

We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices…

We study the maximum number $ex(n,e,H)$ of copies of a graph $H$ in graphs with given number of vertices and edges. We show that for any fixed graph $H$, $ex(n,e,H)$ is asymptotically realized by the quasi-clique provided that the edge…

Let $t$ be an integer such that $t\geq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples $\{a,x_i,y_i\}$, $\{b,x_i,y_i\}$ for $1 \le i \le t$, where the elements $a, b, x_1, x_2, \ldots, x_t,$ $y_1, y_2,…

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations.…

For $r \geq 2$, we show that every maximal $K_{r+1}$-free graph $G$ on $n$ vertices with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$-partite subgraph on $(1 - o(1))n$ vertices. We also show that this is…

A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine…

For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently large…

A long-standing conjecture of Erd\H{o}s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=\Theta(n^r)$. So far this conjecture is known to be true only for rationals of…

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$.…

An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering…

In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on…

In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\le k <r<n$, a $k$-uniform hypergraph…

The celebrated Erd\H{o}s-Hajnal conjecture says that any graph without a fixed induced subgraph $H$ contains a very large homogeneous set. A direct analog of this conjecture is not true for hypergraphs. In this paper we present two natural…

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…