Related papers: On circles enclosing many points

We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider…

Let $C_1, \dots, C_n$ denote the $1/n-$neighborhood of $n$ great circles on $\mathbb{S}^2$. We are interested in how much these areas have to overlap and prove the sharp bounds $$ \sum_{i, j = 1 \atop i \neq j}^{n}{|C_i \cap C_j|^s}…

Two related questions are discussed. The first is when reflection symmetry in a finite set of $i$-dimensional subspaces, $i\in \{1,\dots,n-1\}$, implies full rotational symmetry, i.e., the closure of the group generated by the reflections…

Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice…

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…

Let $P$ be a set of $n$ green and $n - k$ red points in $\mathbb{C}^2$. A line determined by $i$ green and $j$ red points such that $i + j \ge 2$ and $|i - j| \le r$ is called \emph{r-equichromatic}. We establish lower bounds for…

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so…

Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in…

Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there is a vertex disjoint red and blue cycle which span $V(K_n)$. \L uczak, R\"odl, and Szemer\'edi proved Lehel's conjecture for large $n$, Allen gave a different proof…

For a graph whose vertex set is a finite set of points in the Euclidean $d$-space consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect.…

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for…

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting…

The famous Erd\H{o}s-S\'os conjecture states that every graph of average degree more than $t-1$ must contain every tree on $t+1$ vertices. In this paper, we study a spectral version of this conjecture. For $n>k$, let $S_{n,k}$ be the join…

We present an edge labeling of order-$k$ Voronoi diagrams, $V_k(S)$, of point sets $S$ in the plane, and study properties of the regions defined by them. Among them, we show that $V_k(S)$ has a small orientable cycle and path double cover,…

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C in R^d of cardinality (d+1)t, partitioned into t-point subsets C_1,C_2,...,C_{d+1} (which we think of as color classes; e.g., the…

Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if no line generated by any two points in $A$ intersects the convex hull of $B$, and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and…

A covering path for a planar point set is a path drawn in the plane with straight-line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let $\pi(n)$ be the minimum number such…

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value,…

Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $\Omega\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to…