6 papers · 1 filter
Coloring Geometric Hypergraphs: A Survey
Gábor Damásdi, Balázs Keszegh, János Pach +2
The \emph{chromatic number} of a hypergraph is the smallest number of colors needed to color the vertices such that no edge of at least two vertices is monochromatic. Given a famil…
Non-dissective coverings by planks
Andrey Kupavskii, Janos Pach
A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed as the converse of Tarski's plank problem (Bang's theorem): I…
Erdős's unit distance problem and rigidity
János Pach, Orit E. Raz, József Solymosi
According to a classical result of Spencer, Szemerédi, and Trotter (1984), the maximum number of times the unit distance can occur among points in the plane is . Th…
Covering Complete Geometric Graphs by Monotone Paths
Adrian Dumitrescu, János Pach, Morteza Saghafian +1
Given a set of points (vertices) in general position in the plane, the \emph{complete geometric graph} consists of all segments (edges) between the…
On the number of edges of restricted matchstick graphs
Panna Gehér, János Pach, Konrad Swanepoel +1
A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat coun…
Order-forcing in Neural Codes
R. Amzi Jeffs, Caitlin Lienkaemper, Nora Youngs
Convex neural codes are subsets of the Boolean lattice that record the intersection patterns of convex sets in Euclidean space. Much work in recent years has focused on finding com…