215 problems

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

Let $r,d$ be positive integers, and let a Tverberg partition of a set of points be a partition into $r$ parts whose convex hulls intersect. For a set of $(r-1)(d+1)+1$ points in…

Let $\ell_1$ and $\ell_2$ be lines, and let $\mathcal P_1\subseteq\ell_1$ and $\mathcal P_2\subseteq\ell_2$ be sets of $n$ points. For point sets…

Let $r,d$ be positive integers. Given $d+1$ sets $X_1, \ldots, X_{d+1}$, each consisting of $r$ points of $\mathds{R}^d$, consider partitions of their union into $r$ sets…

Let $X$ denote the set of crossing points and $T$ the set of tangency points in a family of $n$ closed curves such that every pair of curves intersects. Richter–Thomassen conjectur…

Let $P\subset\mathbb{R}^d$ be a lattice polytope of dimension $d$ and degree $s$. A Cayley polytope is a lattice polytope expressible as a Cayley sum of lattice polytopes. Cayley c…

Let a family of planar curves be precisely 1-intersecting if every pair of curves has precisely one common point, either a crossing or a tangency, and assume that no three curves s…

Let $D(n)$ be the maximum number of lattice points that can be selected from the $n\times n$ square grid so that no three selected points lie on a common line. An obvious upper bou…

Let $X\subset\mathbb{R}^2$. The invisibility graph $G_X$ of $X$ has vertex set $X$, with two points adjacent when they do not see each other through $X$, and let $\chi(X)$ denote i…

Let $n(d,r)$ be the smallest integer such that, for every map $f:\Delta_{n-1}\rightarrow\mathbb{R}^{d}$ and every coloring of the vertices of $\Delta_{n-1}$ by $d+1$ colors, with e…

Let $r$ and $d$ be positive integers, and let $N$ denote the number of points under consideration. Birch's conjecture. Any $(r-1)(d+1)+1$ points in $\mathbb{R}^{d}$ can be partitio…

Let ${\cal A}$ be a simple arrangement of $n\ge 3$ pairwise intersecting pseudocircles, and let $p_3({\cal A})$ denote the number of triangular cells. Triangle lower-bound conjectu…

Let ${\cal A}$ be a simple digon-free arrangement of $n$ pairwise intersecting circles, and let $p_3({\cal A})$ denote the number of triangular cells. Weak Grünbaum triangle conjec…

Let ${\cal A}$ be a simple digon-free arrangement of $n$ pairwise intersecting pseudocircles, and let $p_3({\cal A})$ denote the number of triangular cells. Grünbaum's triangle con…

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 $r$ be a positive integer. Given $r$ blue, $r$ red, and $r$ green lines in the plane in general position, where a family is in general position when the normal vectors of any t…

Normalize the chessboard to the unit square $U=[0,1]^2$. A configuration consists of white and black queen regions, with no queen of one color attacking a queen of the opposite col…

Erdős–Lovász–Vesztergombi conjecture. The number of distinct such distances is linear in $n$.

Let $d$ be an even integer with $d\ge 6$. A $d$-polytope is periodically-cyclic if its vertices have the periodically-cyclic structure described in the preceding construction, and…

Symmetric six-tuple conjecture. All symmetric six-tuples $(a,b,c,a,b,c)$ are solutions of Problem 1.

Let $P$ be a finite set of points in the plane in general position, meaning that no three points of $P$ are collinear. Jamison's conjecture. The point set $P$ has a spanning path w…

Spherical Erdős distance conjecture. On $S^2$, the Erdős distance conjecture should hold without the logarithmic factor; that is, $A$ should determine at least a constant times $N$…

Let $g(n)$ be the maximum number of facets of a 0/1 polytope in $\mathbb{R}^n$. Facet-growth conjecture. The quantity $g(n)$ is of order $n^{n/2}$ as $n\to\infty$. The paper proves…

Let $t(n)$ be the minimum number of triangulations of an $n$-point set in general position in the plane, and let $T(n)$ be the maximum. A double circle is a convex $k$-gon, with…

Directional combinatorial curvature conjecture. This type of curvature information is encoded in direction-dependent combinatorial quantities of the three-cells: any statistically…