Related papers: Intersecting extremal constructions in Ryser's Con…

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing…

Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is…

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually…

In a recent work, Keusch proved the so-called 1-2-3 Conjecture, raised by Karo\'nski, {\L}uczak, and Thomason in 2004: for every connected graph different from $K_2$, we can assign labels~$1,2,3$ to the edges so that no two adjacent…

Given a graph $H$, the $k$-colored Gallai Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a…

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique…

It is well known that Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. As a generalization, it is natural to ask for the maximum number of copies of…

A transversal coalition in a hypergraph $H$ is a partition of the vertex set $U$ into two subsets $U_1$ and $U_2$ such that neither $U_1$ nor $U_2$ alone intersects every hyperedge of $H$, but their union, $U_1 \cup U_2$, intersects every…

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing…

Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\dots,H_r)$ of graphs if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i \in [r]$. A famous conjecture of Kohayakawa and…

The 1-2-3 Conjecture, posed by Karo\'{n}ski, {\L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The…

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…

We give the first exact and stability results for a hypergraph Tur\'{a}n problem with infinitely many extremal constructions that are far from each other in edit-distance. This includes an example of triple systems with Tur\'{a}n density…

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with $1,2,3$ so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in…

Let $r,s,t\geq2$ be integers. For $r$-graphs $G$ and $F_1,\dots,F_s$, we write $G\to(F_1,\dots,F_s)$ if every $s$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$-th color for some $1\leq i\leq s$. Let…

We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distant at most $r$ in a graph are called $r$-adjacent. The least number of colours…

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of…

We solve the existence problem for $F$-designs for arbitrary $r$-uniform hypergraphs $F$. In particular, this shows that, given any $r$-uniform hypergraph $F$, the trivially necessary divisibility conditions are sufficient to guarantee a…

It follows from known results that every regular tripartite hypergraph of positive degree, with $n$ vertices in each class, has matching number at least $n/2$. This bound is best possible, and the extremal configuration is unique. Here we…

One of the cornerstones of extremal graph theory is a result of F\"uredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a…