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

**Side Note:** If there are more standard definitions for any of the ideas presented here, please let me know.

**Definition:** Say that a subset $S$ of the projective plane is *octahedral* if all lines in $S$ pass through the closure of two opposite faces of a regular octahedron centered at the origin.

**Definition:** Say that a subset $S$ of the projective plane is *weakly octahedral* if every set $S'\subseteq S$ such that $|S'|=3$ is octahedral.

**Equivalent Definition:** Say that 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 $\mathbb{R}^3$.

Now, here is my question.

**Question:** 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 is it necessarily true that each $S_i$ is octahedral?

**Note:** The fact that $S_1,S_2,S_3$ and $S_4$ partition the projective plane seems to be important. I believe that this 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 I was working on a graph colouring problem where the vertex set is the projective plane. For the specific problem, follow this link: http://www.openproblemgarden.org/?q=op/circular_colouring_the_orthogonality_graph

I've also posted the problem to the Open Problem Garden, as the answer seems to be unknown: http://www.openproblemgarden.org/op/partitioning_the_projective_plane