Related papers: Intersecting and $2$-intersecting hypergraphs with…

The following very natural problem was raised by Chung and Erd\H{o}s in the early 80's and has since been repeated a number of times. What is the minimum of the Tur\'an number $\text{ex}(n,\mathcal{H})$ among all $r$-graphs $\mathcal{H}$…

A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been…

We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r-4 for all but finitely many r. This answers a question of Abu-Khazneh, Barat, Pokrovskiy and Szabo, and shows that a long-standing unsolved…

In this paper we consider a natural extremal graph theoretic problem of topological sort, concerning the minimization of the (topological) connectedness of the independence complex of graphs in terms of its dimension. We observe that the…

Erd\H{o}s and S\'os initiated the study of the maximum size of a $k$-uniform set system, for $k \geq 4$, with no singleton intersections $50$ years ago. In this work, we investigate the dual problem: finding the minimum size of a…

In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N ,…

Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply K\"onig's Theorem and every bipartite graph is a…

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)…

A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining…

We consider the Erd{\H{o}}s - Faber - Lov\'{a}sz (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of $r$ regular linear hypergraphs $\textbf{H}$ of size $n$. If $r \ge 4$, $\chi (\textbf{H}) \le…

A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each vertex of $H$ at most $t$ times. We consider the existence of shallow hitting edge sets…

In 1965 Erd\H{o}s conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative…

Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. Erd\H{o}s and Lov\'asz proved that $ \lfloor k! (e-1) \rfloor \leq r(k) \leq k^k.$ Frankl, Ota, and Tokushige improved the lower…

Let $G$ be a bipartite graph without loops and multiple edges on $v\ge 4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne 6$…

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…

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…

Motivated by the well-known conjecture of Ryser which relates maximum matchings to minimum vertex covers in $r$-partite $r$-uniform hypergraphs, Lov\'asz formulated a stronger conjecture. It states that one can always reduce the matching…

An $r$-segment hypergraph $H$ is a hypergraph whose edges consist of $r$ consecutive integer points on line segments in $\mathbb{R}^2$. In this paper, we bound the chromatic number $\chi(H)$ and covering number $\tau(H)$ of hypergraphs in…

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, \eta,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs…

In 1975 Lov\'{a}sz conjectured that every $r$-partite, $r$-uniform hypergraph contains $r-1$ vertices whose deletion reduces the matching number. If true, this statement would imply a well-known conjecture of Ryser from 1971, which states…