Related papers: A new progress on Weak Dirac conjecture

Erd\H{o}s asked the following question: given $n$ points in the plane in almost general position (no 4 collinear), how large a set can we guarantee to find that is in general position (no 3 collinear)? F\"uredi constructed a set of $n$…

Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is…

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the…

It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller…

Let $S$ be a set of $n\geq 7$ points in the plane, no three of which are collinear. Suppose that $S$ determines $n+1$ directions. That is to say, the segments whose endpoints are in $S$ form $n+1$ distinct slopes. We prove that $S$ is, up…

Given a finite point set $P$ in the plane, a subset $S \subseteq P$ is called an island in $P$ if $conv(S) \cap P = S$. We say that $S\subset P$ is a visible island if the points in $S$ are pairwise visible and $S$ is an island in $P$. The…

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a…

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…

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$,…

How many points can be placed in an $n\times n$ grid so that every (affine) line contains at most $k$ points? We prove that for $n \ge k \ge 10^{37}$ the maximum number of points is exactly $kn$. Our proof builds on the recent work of…

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer\'edi and…

Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(\alpha^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p :…

We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an $S_n$-representation. The cases $n = 5, 6$…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex…

We prove that in every direction in the Euclidean plane, there exists a line containing no double exponential time random (ee-random) points. This means each point on these lines has an algorithmically predictable location, to the extent…

Let $P$ be a set of $n$ points in the plane, each point moving along a given trajectory. A {\em $k$-collinearity} is a pair $(L,t)$ of a line $L$ and a time $t$ such that $L$ contains at least $k$ points at time $t$, the points along $L$ do…

We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. Further, we prove…

We prove that for every k, there exists $c_k>0$ such that every graph G on n vertices not inducing a path $P_k$ and its complement contains a clique or a stable set of size $n^{c_k}$.