Related papers: Colourings without monochromatic disjoint pairs

We say that a family of $k$-subsets of an $n$-element set is intersecting, if any two of its sets intersect. In this paper we study different extremal properties of intersecting families, as well as the structure of large intersecting…

The Erd\H{o}s--Ko--Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes,…

We prove that for every $d\in \mathbb{N}$ and a graph class of bounded expansion $\mathscr{C}$, there exists some $c\in \mathbb{N}$ so that every graph from $\mathscr{C}$ admits a proper coloring with at most $c$ colors satisfying the…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a…

Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets. The Erd\H{o}s Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum…

Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson…

Let $M_k$ be a $2n$-vertex graph with $n$ pairwise disjoint edges and let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that span exactly $p$ edges and $s$ isolated vertices. We prove that for $n\ge 2p+s$ this family has the…

In 1981, Erd\H{o}s and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let…

We formulate a geometric version of the Erd\H{o}s-Hajnal conjecture that applies to finite projective geometries rather than graphs, in both its usual 'induced' form and the multicoloured form. The multicoloured conjecture states, roughly,…

In the 1960s, Erd\H{o}s and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by…

The optimality of the Erd\H{o}s-Rado theorem for pairs is witnessed by the colouring $\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$ recording the least point of disagreement between two functions. This colouring has no monochromatic…

We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of $\mathbb{F}_3^n$ as well as obtaining the first non-trivial lower bound…

The celebrated {Erd\H{o}s-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for…

A chamber of the vector space $\mathbb{F}_q^n$ is a set $\{S_1,\dots,S_{n-1}\}$ of subspaces of $\mathbb{F}_q^n$ where $S_1\subset S_2\subset \dotso \subset S_{n-1}$ and $\dim(S_i)=i$ for $i=1,\dots,n-1$. By $\Gamma_n(q)$ we denote the…

The celebrated Erd\H{o}s-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona,…

We address several related problems on combinatorial discrepancy of trees in a setting introduced by Erd\H{o}s, F\"{u}redi, Loebl and S\'{o}s. Given a fixed tree $T$ on $n$ vertices and an edge-colouring of the complete graph $K_n$, for…

Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erd\H{o}s-Rado sunflower lemma.…

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich…