Partitioning the Projective Plane

Importance: Medium ✭✭
Author(s): Noel, Jonathan A.
Subject: Geometry
Recomm. for undergrads: yes
Posted by: Jon Noel
on: August 27th, 2013

Throughout this post, by \emph{projective plane} we mean the set of all lines through the origin in $\mathbb{R}^3$.

\begin{definition} Say that a subset $S$ of the projective plane is \emph{octahedral} if all lines in $S$ pass through the closure of two opposite faces of a regular octahedron centered at the origin. \end{definition}

\begin{definition} Say that a subset $S$ of the projective plane is \emph{weakly octahedral} if every set $S'\subseteq S$ such that $|S'|=3$ is octahedral. \end{definition}

\begin{conjecture} Suppose that the projective plane can be partitioned into four sets, say $S_1,S_2,S_3$ and $S_4$ such that each set $S_i$ is weakly octahedral. Then each $S_i$ is octahedral. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

Also, see the \href[posting]{http://mathoverflow.net/questions/140413/partitioning-the-projective-plane} on mathoverflow.

There is an equivalent definition of the "weakly octahedral" condition which may be useful.

\begin{lemma} A subset $S$ of the projective plane is weakly octahedral if for any three lines in $S$ and any three vectors $x, y$ and $z$ which span these lines, we have $$\langle x,y\rangle \cdot\langle x,z\rangle \cdot\langle y,z\rangle \geq 0$$ where $\langle\cdot,\cdot\rangle$ is the standard (dot) inner product on $\mathbb{R}^3$. \end{lemma}

The fact that $S_1,S_2,S_3$ and $S_4$ partition the projective plane seems to be important. Here is an example of a weakly octahedral set that is not octahedral: Fix any vector $x$ and let $S$ be the set of all lines which are spanned by vectors which meet $x$ at an angle strictly less than $\frac{\pi}{4}$.

This question came up while working on another problem posted to this site: \href[Circular colouring the orthogonality graph]{http://www.openproblemgarden.org/?q=op/circular_colouring_the_orthogonality_graph}. It is possible that a solution to the problem stated here can be applied to solve this problem. Moreover, it may be useful in proving that the real orthogonality graph (defined in the other \href[posting]{http://www.openproblemgarden.org/?q=op/circular_colouring_the_orthogonality_graph}) has (essentially) only one proper $4$-colouring.

Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)


* indicates original appearance(s) of problem.