Related papers: Petruska's question on planar convex sets

If K' and K are convex bodies of the plane such that K' is a subset of K then the perimeter of K' is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with…

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more…

In 1935, Erd\H{o}s and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a…

Let $f(n,k)$ be the minimum number of edges that must be removed from some complete geometric graph $G$ on $n$ points, so that there exists a tree on $k$ vertices that is no longer a planar subgraph of $G$. In this paper we show that…

We prove that every finite family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by $9$ points. This improves the bound of $13$ proved by Gy\'arf\'as, Kleitman, and T\'oth in 2001.

A (4,5,6)-fullerene is a plane cubic graph whose faces are only quadrilaterals, pentagons and hexagons, which includes all (4,6)- and (5,6)-fullerenes. A connected graph $G$ with at least $2k+2$ vertices is $k$-extendable if $G$ has perfect…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes.…

Let $\F$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by $\F$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in…

Let $ES_{d}(n)$ be the smallest integer such that any set of $ES_{d}(n)$ points in $\mathbb{R}^{d}$ in general position contains $n$ points in convex position. In 1960, Erd\H{o}s and Szekeres showed that $ES_{2}(n) \geq 2^{n-2} + 1$ holds,…

We study the maximum number of straight-line segments connecting $n$ points in convex position in the plane, so that each segment intersects at most $k$ others. This question can also be framed as the maximum number of edges of an outer…

Say that a subset S of the plane is a "circle-center set" if S is not a subset of a line, and whenever we choose three noncollinear points from S, the center of the unique circle through those three points is also an element of S. A problem…

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has…

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…

In this paper we study the maximum number of hyperedges which may be in an $r$-uniform hypergraph under the restriction that no pair of vertices has more than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the even-cycle…

We consider the Tarski--Bang problem about covering of convex bodies by planks. The results of this kind give a lower bound on the sum of widths of planks (regions between a pair of parallel hyperplanes) covering a given convex body.…

We consider colored variants of a class of geometric-combinatorial questions on $k$-gons and empty $k$-gons that have been started around 1935 by Erd\H{o}s and Szekeres. In our setting we have $n$ points in general position in the plane,…

An untouchable set in a projective plane is a set of points such that no line of the plane meets the set in exactly one point. Recently, H\'eger and Nagy (Avoiding Secants of Given Size in Finite Projective Planes, J. Combin. Des.…

In this short paper, Kuratowski problem will be investigated in vector space. The highest number of distinct sets that can be generated from one convex set in linear space by repeatedly applying algebraic closure and complement in any order…

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\] where $\delta >0$ is…

A family of sets has the $(p, q)$ property if among any $p$ members of it some $q$ intersect. It is shown that if a finite family of compact convex sets in $\R^2$ has the $(p+1,2)$ property then it is pierced by $\lfloor \frac{p}{2} \rfloor…