Related papers: A hierarchy of maximal intersecting triple systems

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices. In this…

We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_{\ell}$ denote the path on $\ell$ vertices, $S_{\ell-1}$ denote the star on $\ell$…

The Tur\'{a}n number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Tur\'{a}n numbers for a path of arbitrary…

An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is…

We prove the following the generalized Tur\'an type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in…

A k-uniform linear cycle of length s is a cyclic list of k-sets A_1,..., A_s such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k at least 5 and s at least 3 and sufficiently large n we…

The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_k$ denote the path on $k$ vertices, $S_k$ denote the star on $k+1$ vertices and…

For two graphs $J$ and $H$, the generalized Tur\'{a}n number, denoted by $ex(n,J,H)$, is the maximum number of copies of $J$ in an $H$-free graph of order $n$. A linear forest $F$ is the disjoint union of paths. In this paper, we determine…

We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new…

For a compact surface $ S $ with a finite set of marked points $ P $, we define a 1-system to be a collection of arcs which are pairwise non-homotopic and intersect pairwise at most once. We prove that, up to equivalence, there are exactly…

In a recent paper (2024) M. Buratti and M.E:Muzychuck have established some lower bounds on the number of non isomorphic cyclic Steiner Triple Systems of order $v\equiv 1$ (mod $6$). We complete their result to the case $v\equiv 3$ (mod…

A perfect matching in a 3-uniform hypergraph on $n=3k$ vertices is a subset of $\frac{n}{3}$ disjoint edges. We prove that if $H$ is a 3-uniform hypergraph on $n=3k$ vertices such that every vertex belongs to at least ${n-1\choose 2} -…

Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is…

We show that every sufficiently large plane triangulation has a large collection of nested cycles that either are pairwise disjoint, or pairwise intersect in exactly one vertex, or pairwise intersect in exactly two vertices. We apply this…

A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent…

Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $\delta_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of…

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$…

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect…