Related papers: Separating hypergraph Tur\'an densities

Chao and Yu introduced an entropy method for hypergraph Tur\'an problems, and used it to show that the family of $\lfloor k/2\rfloor$ $k$-uniform tents have Tur\'an density $k!/k^k$. Il'kovi\v{c} and Yan improved this by reducing to a…

In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many…

Given positive integers k and r where 4 divides k and k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who…

Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding with the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal was to…

A tight $\ell$-cycle minus an edge $C_\ell^-$ is the $3$-graph on the vertex set $[\ell]$, where any three consecutive vertices in the string $123\ldots\ell 1$ form an edge. We show that for every $\ell\ge 5$, $\ell$ not divisible by $3$,…

Given graphs $H$ and $F$, the generalized Tur\'an number $\ex(n, H, F)$ is defined as the maximum number of copies of $H$ in an $n$-vertex graph that contains no copy of $F$. The suspension $\widehat{F}$ of a graph $F$ is obtained by adding…

Let $P$ be a set of $n$ points in the plane, and let $\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and…

Combinatorial batch codes provide a tool for distributed data storage, with the feature of keeping privacy during information retrieval. Recently, Balachandran and Bhattacharya observed that the problem of constructing such uniform codes in…

As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…

A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\cal{F}}$ of hypergraphs, the linear Tur\'an number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform…

A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the…

Let $F$ be a strictly balanced $r$-uniform hypergraph with $e>2$ edges and $r$-density $m$. We give a new short proof of the fact that the Tur\'an number $\ex(n, F)$ is greater than $c\, n^{r-1/m} (\log n)^{1/(e-1)}$ where $c$ depends only…

For a graph $H$ and a $k$-chromatic graph $F,$ if the Tur\'an graph $T_{k-1}(n)$ has the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough), then $H$ is called $F$-Tur\'an-good, or $k$-Tur\'an-good…

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…

The $r$-expansion of a $k$-uniform hypergraph $H$, denoted by $H^{(+r)}$, is an $r$-uniform hypergraph obtained by enlarging each $k$-edge of $H$ with a set of $r-k$ vertices of degree one. The random Tur\'an number…

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…

Degeneracy plays an important role in understanding Tur\'an- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of…

Given a family of $3$-graphs $\mathcal{F}$, the uniform Tur\'{a}n density $\pi_{\therefore}(\mathcal{F})$ is defined as the infimum $d\in[0,1]$ for which any sufficiently large uniformly $d$-dense $3$-graph - that is, a $3$-graph which has…

Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$, and let $s(F):=\sup\{s: \exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\}$. Following recent work of…

We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced $k$-clique and a graph in which all k-subgraphs have density at most $1-\epsilon$, requires $n^{\tilde \Omega(log n)}$…