22 problems

Quadratic-hole conjecture. The quadratic upper bounds for the minimum numbers of 5-holes and 6-holes are tight:

Big-line Big-clique Conjecture. For any $k\geq 3$ and $\ell\geq 3$, there is an integer $n=n(k,\ell)$ such that every finite set of at least $n$ points in the plane contains either…

Let $P$ be a finite set of points in the plane. Two distinct points $v,w\in P$ are visible with respect to $P$ if no point of $P$ lies in the open line segment $\overline{vw}$. For…

For positive integers $v\ge 3$ and $i\le v$, a point set is in almost convex position with parameters $(v,i)$ if it consists of the vertices of a convex $v$-gon together with $i$ i…

Let $A$ be a point set in general position in the plane, let $A_I$ be its set of interior points, and for each $W\subseteq A_I$ let $\mathcal{PT}_{W}(A)$ denote the pseudo-triangul…

Let $f_S(n)$ denote the maximum number of subsets congruent to a finite planar pattern $S$ among $n$ points in the plane. In particular, let $f_\Delta(n)$ be the corresponding quan…

Monotonicity conjecture. For all integers $n>k\ge 6$,

Exponent conjecture for $h_k(n)$. For every fixed integer $k\ge 6$,

Benjamini and Tzalik's conjecture. There exists a reconstructible subset of $V$ of size $m/n$.

Giro, Illingworth, Michel, Powierski, and Scott's conjecture. For every fixed $\varepsilon>0$, when $p=(1+\varepsilon)/n$, one can reconstruct a subset of $V$ of size linear in $n$…

Neighborhood-island conjecture. The neighborhood of some point $p\in P$ contains an island of size $f(n)$, where $f(n)$ tends to infinity as $n$ tends to infinity.

Horton-set order-type conjecture. For each $n,ell itypemathbb{N}$, every sufficiently large $ell$-hole-free set in general position in $mathbb{R}^2$ contains an $n$-point subset wh…

Let $S$ be a finite set of points in the plane in general position, and let $k\geq 3$ and $\ell\geq 1$ be integers. For a convex $k$-gon determined by points of $S$, consider the n…

Let $P$ be a set of $n$ points in the plane such that $D(P)=O(n/\sqrt{\log n})$, where $D(P)$ is the number of distinct distances determined by pairs of points of $P$. Erdős conjec…

For a positive integer $c$, let $M_3(c,s)$ be the least integer such that every set of at least $M_3(c,s)$ points in the plane, with no three collinear and colored with $c$ colors,…

Let $g_\mathrm{col}(n)$ denote the relevant extremal quantity for colored radial orderings of bichromatic sets of $2n$ points. Quadratic colored-ordering conjecture. … The paper no…

Let $S$ be a set of $n$ points in general position in the plane, meaning that no three points are collinear. General-position quartic lower-bound conjecture. $S$ has at least…

Let $g(n)$ denote the minimum number of distinct radial orderings determined by a set of $n$ points in strong general position in the plane. Quartic growth conjecture. … The paper…

Let $P$ be a set of $n$ points in the plane in general position, meaning that no three are collinear and no four are cocircular. A pair of points has the required circle depth when…

Let $P\subset\mathbb{R}^3$ be a set of $n$ points in convex position. The depth of a segment is the minimum number of points that must be removed from $P$ so that the segment is no…

Let $P$subset$\mathbb{R}^2$ be a set of $n$ points in general position, meaning that no three are collinear and no four are cocircular. For a pair of points $p,q\in P$, consider th…

Let $S$ be a finite point set in $cmathbb{R}^d$. A simplicial half-net is a family of simplices determined by points of $S$ such that every halfspace containing at least half the p…