97 problems

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 $P\subset\mathbb{R}^d$ 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…

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…

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 $F_1,\ldots,F_{d+1}$ be families of $r$ points each in $\mathds{R}^d$, considered as color classes. A colorful partition is a partition into $r$ sets $A_1,\ldots,A_r$ such that…

Let $k$ be a fixed positive integer, and let a family consist of $n$ pairwise intersecting bi-infinite $x$-monotone curves such that any two curves intersect at most $k$ times. Lin…

Let $A$ be a dense $n$-element point set in general position in the plane, and let $K_n[A]$ denote the complete geometric graph induced by $A$. A path is monotone if its edge set i…

Let $S$ be an even set of points in the plane, and let $\mathcal M$ be a max-sum matching of $S$. Define $t(S)$ as the minimum, over points $y$ in the plane, of…

Let $S$ be an even set of points in the plane, and let $\{a_ib_i:i=1,\dots,n\}$ be a max-sum matching of $S$, where $|S|=2n$. A point $o$ is required to satisfy, for every matched…

Let $L_{\Delta}=\left\{\left(a+\frac{1}{2}b,\frac{\sqrt{3}}{2}b\right):a,b\in\mathbb{Z}\right\}$ be the triangular lattice. A configuration is $k$-optimal if it has the maximum pos…

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

Middle-regime doubling conjecture. For every $m\ge4$ and every $n$ in this middle regime,

Tree--necklace Gauss-load conjecture. For tree-like and tree--necklace shadows, the minimum Gauss load is obtained by a reduced tangent-angle realization of a coorientation minimiz…

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 $n$ blue points, no three on a line, and $n$ red points disjoint from the blue points be given. Assume that every line through two blue points contains a red point. Milićević's…

Let $P$ be a non-collinear set of $n$ two-colored points, and suppose that no line contains more than $(1/2-\delta)n$ points, where $\delta>0$. Böröczky extremality conjecture. For e…

Blue-polygon bichromatic-face conjecture. Given a blue $k$-gon $P$ with at least $k-4$ red pseudolines passing through, there exists a bichromatic triangle or quadrangle on its ins…

Triangle-pseudoline incidence graph conjecture. The triangle-pseudoline incidence graph of a pseudoline arrangement is connected.

Björner–Las Vergnas–Sturmfels–White–Ziegler conjecture. Every bicolored arrangement has a bichromatic triangle.

Central-slice extremality conjecture. For every centrally symmetric $k$-dimensional polytope $P$, the upper bound on the number of vertices of a $(k-1)$-dimensional slice of $P$ is…

Generic central slice conjecture. The upper bound on the number of vertices of a $k$-dimensional slice of $C_d$ can be attained by a generic central $k$-dimensional slice for every…

Let an $m$-distance set be a set of points whose pairwise distances take at most $m$ values. The multiplicity of a distance is the number of unordered pairs of points in the set re…

Let a triangular lattice be the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. Consider a regular hexagon or an equiangular hexagon w…