10 problems

Big-Line-Big-Clique Conjecture. For all integers $t$ and $\ell$ there is an integer $n$ such that every finite set $P$ of at least $n$ points in the plane either contains $\ell$ co…

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

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.

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…

Big line or big clique conjecture. For all integers $k$ and $\ell$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains either $\ell$ col…

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 $S$ 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…