# Partitioning

## Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by *projective plane* we mean the set of all lines through the origin in .

**Definition**Say that a subset of the projective plane is

*octahedral*if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.

**Definition**Say that a subset of the projective plane is

*weakly octahedral*if every set such that is octahedral.

**Conjecture**Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.

Keywords: Partitioning; projective plane

## 2-colouring a graph without a monochromatic maximum clique ★★

**Conjecture**If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.

Keywords: maximum clique; Partitioning

## Convex 'Fair' Partitions Of Convex Polygons ★★

Author(s): Nandakumar; Ramana

**Basic Question:** Given any positive integer *n*, can any convex polygon be partitioned into *n* convex pieces so that all pieces have the same area and same perimeter?

**Definitions:** Define a *Fair Partition* of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a *Convex Fair Partition*.

**Questions:** 1. (Rephrasing the above 'basic' question) Given any positive integer *n*, can any convex polygon be convex fair partitioned into n pieces?

2. If the answer to the above is *"Not always''*, how does one decide the possibility of such a partition for a given convex polygon and a given *n*? And if fair convex partition is allowed by a specific convex polygon for a give *n*, how does one find the *optimal* convex fair partition that *minimizes* the total length of the cut segments?

3. Finally, what could one say about *higher dimensional analogs* of this question?

**Conjecture:** The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: *Every* convex polygon allows a convex fair partition into *n* pieces for any *n*

Keywords: Convex Polygons; Partitioning