Related papers: An Erd\H{o}s-Gallai type theorem for uniform hyper…

We study the structure of $r$-uniform hypergraphs containing no Berge cycles of length at least $k$ for $k \leq r$, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such…

In 1973 P. Erd\H{o}s and L. Lov\'asz noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erd\H{o}s and Lov\'asz proved that…

A classical result of Erd\H{o}s and Hajnal claims that for any integers $k, r, g \geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ with chromatic number at least $k$. This implies that there are sparse hypergraphs such that…

A theorem of Shearer states that every $n$-vertex triangle-free graph of maximum degree $d \geq 2$ contains an independent set of size at least $(d\log d - d + 1)/(d - 1)^2 \cdot n$. Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di…

The well-known Erd\H{o}s-Gallai Theorem gave the Tur\'an number of paths. Bushaw and Kettle generalized this result to consider the Tur\'an number of disjoint paths. Since then, many studies are focused on the Tur\'an number of linear…

An $r$-uniform linear cycle of length $\ell$, denoted by $C^r_{\ell}$, is an $r$-graph with $\ell$ edges $e_1,e_2,\dots,e_{\ell}$ where $e_i=\{v_{(r-1)(i-1)},v_{(r-1)(i-1)+1},\dots,v_{(r-1)i}\}$ (here $v_0=v_{(r-1)\ell}$). For $0<\delta<1$…

A graph $\Gamma$ is said to be universal for a class of graphs $\mathcal{H}$ if $\Gamma$ contains a copy of every $H \in \mathcal{H}$ as a subgraph. The number of edges required for a host graph $\Gamma$ to be universal for the class of…

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…

In this paper, we address the following problem due to Frankl and F\"uredi (1984). What is the maximum number of hyperedges in an $r$-uniform hypergraph with $n$ vertices, such that every set of $r+1$ vertices contains $0$ or exactly $2$…

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with…

In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz [12],…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…

Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\lceil n/2\rceil$ paths. Let $\mathcal{G}_{k}$ be the class of all $2k$-regular graphs of girth at least $2k-2$ that admit a…

Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The…

Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist injections $\psi:U\to V$ and $\varphi:E\to \mathcal{E}$ such that for every $e=\{u,v\}\in E$,…

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all…

Gallai's path decomposition conjecture states that the edges of any connected graph on n vertices can be decomposed into at most (n+1)/2 paths. We confirm that conjecture for all graphs with maximum degree at most five.

A keyring is a graph obtained by appending $r \geq 1$ leaves to one of the vertices of a cycle. We prove that for every $r \leq (k-1)/2$, any graph with average degree more than $k-1$ contains a keyring with $r$ leaves and at least $k$…

Kalai conjectured that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r} {n \choose r-1}$ edges contains all tight $r$-trees of some fixed size $t$. We prove Kalai's conjecture for $r$-partite $r$-uniform hypergraphs.…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…