Related papers: Many collinear k-tuples with no k+1 collinear poin…

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…

The maximum number of non-crossing straight-line perfect matchings that a set of $n$ points in the plane can have is known to be $O(10.0438^n)$ and $\Omega^*(3^n)$. The lower bound, due to Garc\'ia, Noy, and Tejel (2000) is attained by the…

The point sets of all known optimal rectilinear drawings of $K_n$ share an unmistakeable clustering property, the so--called {\em 3--decomposability}. It is widely believed that the underlying point sets of all optimal rectilinear drawings…

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…

Let $G$ be a 3-partite graph with $k$ vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matou\u{s}ek asked for the maximum number of triangles in such a graph. A simple construction…

In 2016, Dowden initiated the study of planar Tur\'an-type problems, which has since attracted considerable attention. Recently, Bekos et al. proved that every $K_3$-free $1$-planar graph on $n\ge 4$ vertices has at most $3n-6$ edges. In…

Chv\'{a}tal and Klincsek (1980) gave an $O(n^3)$-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set $P$ of $n$ points in the plane. This paper examines a generalization of the problem,…

Let $k \geq 1$ be an integer. A set $A \subset \mathbb{Z}$ is a $k$-fold Sidon set if $A$ has only trivial solutions to each equation of the form $c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ where $0 \leq |c_i | \leq k$, and $c_1 + c_2 + c_3…

We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is…

A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are…

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek…

We show that each set of $n\ge 2$ points in the plane in general position has a straight-line matching with at least $(5n+1)/27$ edges whose segments form a connected set, and such a matching can be computed in $O(n \log n)$ time. As an…

A set P of points in R^2 is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n>=15. Conversely,…

We show that any set of $n$ points in general position in the plane determines $n^{1-o(1)}$ pairwise crossing segments. The best previously known lower bound, $\Omega\left(\sqrt n\right)$, was proved more than 25 years ago by Aronov, Erd\H…

We show that the number of partial triangulations of a set of $n$ points on the plane is at least the $(n-2)$-nd Catalan number. This is tight for convex $n$-gons. We also describe all the equality cases.

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite…

P\'or and Wood conjectured that for all $k,l \ge 2$ there exists $n \ge 2$ with the following property: whenever $n$ points, no $l + 1$ of which are collinear, are chosen in the plane and each of them is assigned one of $k$ colours, then…

Let $P$ be a set of $n$ points in the plane that determines at most $n/5$ distinct distances. We show that no line can contain more than $O(n^{43/52}{\rm polylog}(n))$ points of $P$. We also show a similar result for rectangular distances,…

For a set $P$ of $n$ points in $\mathbb R^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…