# combinatorial geometry

## The Double Cap Conjecture ★★

Author(s): Kalai

\begin{conjecture} The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\pi/4$ around the north and south poles. \end{conjecture}

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

## Point sets with no empty pentagon ★

Author(s): Wood

\begin{problem} Classify the point sets with no empty pentagon. \end{problem}

Keywords: combinatorial geometry; visibility graph

## Erdös-Szekeres conjecture ★★★

\begin{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. \end{conjecture}

Keywords: combinatorial geometry; Convex Polygons; ramsey theory