Partitioning the Projective Plane ★★

Author(s): Noel

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}

Keywords: Partitioning; projective plane

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

Author(s): Hoang; McDiarmid

\begin{conjecture} If $G$ is a non-empty graph containing no induced odd cycle of length at least $5$, then there is a $2$-vertex colouring of $G$ in which no maximum clique is monochromatic. \end{conjecture}

Keywords: maximum clique; Partitioning

Convex 'Fair' Partitions Of Convex Polygons ★★

Author(s): Nandakumar; Ramana

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

\textbf{Definitions:} Define a \emph{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 \emph{Convex Fair Partition}.

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

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

3. Finally, what could one say about \emph{higher dimensional analogs} of this question?

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

Keywords: Convex Polygons; Partitioning

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