Unsolved Problems

Suppose $A \subset \mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic cur...

Fix $k$. Let $A \subset \mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\delta n^2$ pairs $(x, y) \in A \tim...

What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a s...

Let $\Gamma$ be a smooth codimension 2 surface in $\mathbb{R}^n$. Must $\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently...

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?...

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?...

Problem Does there exist a dense set $S \subseteq {\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational?...

Question What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices?...

Conjecture If a finite set of unit balls in $\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the...

Conjecture Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon....

If $X \subseteq {\mathbb R}^2$ is a finite set of points which is 2-colored, an empty triangle is a set $T \subseteq X$ with $|T|=3$ so that the conve...

Question Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\ldots,a_n$ inside an $n$-dimensional ...

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 ...

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...

Conjecture Every convex polytope has a non-overlapping edge unfolding....