Point sets with no empty pentagon
Problem Classify the point sets with no empty pentagon....
Covering a square with unit squares
Conjecture For any integer $n \geq 1$, it is impossible to cover a square of side greater than $n$ with $n^2+1$ unit squares....
Convex uniform 5-polytopes
Problem Enumerate all convex uniform 5-polytopes....
Partitioning the Projective Plane
Throughout this post, by projective plane we mean the set of all lines through the origin in $\mathbb{R}^3$. Definition Say that a subset $S$ of the ...
Dirac's Conjecture
Conjecture For every set $P$ of $n$ points in the plane, not all collinear, there is a point in $P$ contained in at least $\frac{n}{2}-c$ lines determ...
General position subsets
Question What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ ...
Generalised Empty Hexagon Conjecture
Conjecture For each $\ell\geq3$ there is an integer $f(\ell)$ such that every set of at least $f(\ell)$ points in the plane contains $\ell$ collinear ...
Chromatic number of associahedron
Conjecture Associahedra have unbounded chromatic number....
Convex Equipartitions with Extreme Perimeter
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total ...
Edge-Unfolding Convex Polyhedra
Conjecture Every convex polyhedron has a (nonoverlapping) edge unfolding....
Jacobian Conjecture
Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\ldots,f_n$ of polynomials in variables $x_1,\ldots,x_n$ defines an automorphi...
The Hodge Conjecture
Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subva...
Fat 4-polytopes
The fatness of a 4-polytope $P$ is defined to be $(f_1 + f_2)/(f_0 + f_3)$ where $f_i$ is the number of faces of $P$ of dimension $i$. Question Does ...
Continous analogue of Hirsch conjecture
Conjecture The order of the largest total curvature of the primal central path over all polytopes defined by $n$ inequalities in dimension $d$ is $n$....
Cube-Simplex conjecture
Conjecture For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is...
Extension complexity of (convex) polygons
The extension complexity of a polytope $P$ is the minimum number $q$ for which there exists a polytope $Q$ with $q$ facets and an affine mapping $\pi$...
Durer's Conjecture
Conjecture Every convex polytope has a non-overlapping edge unfolding....