Related papers: Extremal numbers and Sidorenko's conjecture

A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner, Lemons, Palmer, Patk\'os and Sz\'ecsi conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost…

The Ramsey multiplicity constant of a graph $H$ is the minimum proportion of copies of $H$ in the complete graph which are monochromatic under an edge-coloring of $K_n$ as $n$ goes to infinity. Graphs for which this minimum is…

The Erd\H{o}s-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erd\H{o}s-Simonovits type stability theorem in multi-partite graphs. Different from the Erd\H{o}s-Simonovits stability…

Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection of $m$…

For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much…

In this paper we disprove a conjecture of Lidick\'y and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies…

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the…

A well-known conjecture by Erd\H{o}s states that every triangle-free graph on $n$ vertices can be made bipartite by removing at most $n^2/25$ edges. This conjecture was known for graphs with edge density at least $0.4$ and edge density at…

A famous conjecture of Tuza \cite{tuza} is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal number of edge-disjoint triangles. We propose a wider setting for this conjecture. For a…

For a sequence $(H_i)_{i=1}^k$ of graphs, let $\textrm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$.…

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be…

A sparse version of Mantel's Theorem is that, for sufficiently large $p$, with high probability (w.h.p.), every maximum triangle-free subgraph of $G(n,p)$ is bipartite. DeMarco and Kahn proved this for $p>K \sqrt{\log n/n}$ for some…

For a graph $G$ and a hereditary property $\mathcal{P}$, let $\text{ex}(G,\mathcal{P})$ denote the maximum number of edges of a subgraph of $G$ that belongs to $\mathcal{P}$. We prove that for every non-trivial hereditary property…

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…

Erd\H{o}s and Simonovits asked the following question: For an integer $r\geq 2$ and a family of non-bipartite graphs $\mathcal{H}$, determine the infimum of $\alpha$ such that any $\mathcal{H}$-free $n$-vertex graph with minimum degree at…

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…

A result of Simonovits and S\'os states that for any fixed graph $H$ and any $\epsilon > 0$ there exists $\delta > 0$ such that if $G$ is an $n$-vertex graph with the property that every $S \subseteq V(G)$ contains $p^{e(H)} |S|^{v(H)} \pm…

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 $\mathcal{H}$.…

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…