Related papers: Improved bounds for the Erd\H{o}s-Rogers function

A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…

The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass e and Watrigant in 2020, allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the…

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$,…

A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let…

Let $f(n,v,e)$ denote the maximum number of edges in a $3$-uniform hypergraph not containing $e$ edges spanned by at most $v$ vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of…

A $k$-uniform hypergraph with $n$ vertices is an $(n,k,\ell)$-omitting system if it does not contain two edges whose intersection has size exactly $\ell$. If in addition it does not contain two edges whose intersection has size greater than…

We establish the following strengthening of the celebrated Andr{\'a}sfai--Erd\H{o}s--S\'{o}s theorem: If $G$ is an $n$-vertex $K_{r+1}$-free graph whose minimum degree $\delta(G)$ and maximum degree $\Delta(G)$ satisfy \begin{align*}…

Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called…

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…

Erd\H{o}s asked for the largest size $f(n)$ of a subset of $\{1,\dots,n\}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such…

Let $H=(V,E)$ be an $s$-uniform hypergraph of order $n$ and $k\geq 0$ be an integer. A $k$-independent set $S\subseteq H$ is a set of vertices such that the maximum degree in the hypergraph induced by $S$ is at most $k$. Denoted by…

We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if…

We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or…

In 1974, Erd\H{o}s posed the following problem. Given an oriented graph $H$, determine or estimate the maximum possible number of $H$-free orientations of an $n$-vertex graph. When $H$ is a tournament, the answer was determined precisely…

We give upper bounds for the number $\Phi_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular,…

We obtain several lower bounds on the $\textsf{Max-Cut}$ of $d$-degenerate $H$-free graphs. Let $f(m,d,H)$ denote the smallest $\textsf{Max-Cut}$ of an $H$-free $d$-degenerate graph on $m$ edges. We show that $f(m,d,K_r)\ge…

Let $H$ and $F$ be hypergraphs. We say $H$ contains $F$ as a trace if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ contains a subhypergraph isomorphic to $F$. In this paper we give an upper bound on the…

We consider the problem of estimating the edge density of densest $K$-node subgraphs of an Erd\"os-R\'{e}nyi graph $\mathbb{G}(n,1/2)$. The problem is well-understood in the regime $K=\Theta(\log n)$ and in the regime $K=\Theta(n)$. In the…

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very…

We provide a new lower bound on the number of $(\leq k)$-edges of a set of $n$ points in the plane in general position. We show that for $0 \leq k \leq\lfloor\frac{n-2}{2}\rfloor$ the number of $(\leq k)$-edges is at least $$ E_k(S) \geq…