Related papers: On small $n$-uniform hypergraphs with positive dis…

The \emph{Tur\'an function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs where the key…

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can…

Let $f_r(n,v,e)$ denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices, in which the union of any $e$ distinct edges contains at least $v+1$ vertices. The study of $f_r(n,v,e)$ was initiated by Brown, Erd{\H{o}}s…

The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

Let $G$ be a $d$-regular graph on $n$ vertices. Frieze, Gould, Karo\'nski and Pfender began the study of the following random spanning subgraph model $H=H(G)$. Assign independently to each vertex $v$ of $G$ a uniform random number $x(v) \in…

Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $R\cup B$ does not contain a copy of $H$? Call the largest…

Given $k$-uniform hypergraphs $G$ and $H$ on $n$ vertices with densities $p$ and $q$, their relative discrepancy is defined as $\hbox{disc}(G,H)=\max\big||E(G')\cap E(H')|-pq\binom{n}{k}\big|$, where the maximum ranges over all pairs…

A set $A$ of vertices in an $r$-uniform hypergraph $\mathcal H$ is covered in $\mathcal H$ if there is some vertex $u\not\in A$ such that, for every $(r-1)$-set $B\subset A$, the set $\{u\}\cup B$ is in $\mathcal H$. Erdos and Moser (1970)…

Given graphs $H_1, H_2$, a {red, blue}-coloring of the edges of a graph $G$ is a critical coloring if $G$ has neither a red $H_1$ nor a blue $ H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G$ admits a critical coloring, but…

A mixed hypergraph is a triple $H=(V,\mathcal{C},\mathcal{D})$, where $V$ is a set of vertices, $\mathcal{C}$ and $\mathcal{D}$ are sets of hyperedges. A vertex-coloring of $H$ is proper if $C$-edges are not totally multicolored and…

In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3,…

Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on~$n$ vertices not containing a subgraph with $k$~edges and at most $s$~vertices. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit $$\lim_{n\to…

Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Tur\'an theory. Using results…

An $r$-uniform hypergraph is called $t$-union-free if any two distinct subsets of at most $t$ edges have distinct union. The study of union-free hypergraphs has multiple origins and a long history, dating back to the works of Kautz and…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an r-partite r-uniform…

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…