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Revision difference : finite plane |
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Version 6 |
| Let $\mathcal{H} = (V, \mathcal{E})$ be a linear space. A \emph{finite plane} is an intersecting linear space. That is to say, a linear space in which any two edges in $\mathcal{E}$ have a nonempty intersection. |
Let $\mathcal{H} = (V, \mathcal{E})$ be a linear space. A \emph{finite plane} is an intersecting linear space. That is to say, a linear space in which any two edges in $\mathcal{E}$ have a nonempty intersection. |
| Finite planes are rather restrictive hypergraphs, and the following holds. |
Finite planes are rather restrictive hypergraphs, and the following holds. |
| \begin{thm} |
\begin{thm} |
| Let $\mathcal{H} = (V, \mathcal{E})$ be a finite plane. Then for some positive integer $k$, $\mathcal{H}$ is $(k+1)-$regular, $(k+1)-$uniform, and $|\mathcal{E}| = |V| = k^2 + k + 1$. |
Let $\mathcal{H} = (V, \mathcal{E})$ be a finite plane. Then for some positive integer $k$, $\mathcal{H}$ is $(k+1)-$regular, $(k+1)-$uniform, and $|\mathcal{E}| = |V| = k^2 + k + 1$. |
| \end{thm} |
\end{thm} |
| The above $k$ is the \emph{\PMLinkEscape{order}} of the finite plane. It is not known in general if finite planes exist of \PMLinkEscape{order} other than $k$ a power of a prime. The terminology "finite plane" is suggestive, as we can think of the edges as a finite collection of lines in Euclidean space in general position so that they all intersect pairwise in exactly one point. The added restriction that all pairs of vertices determine an edge (a line), however, makes it impossible to depict a finite plane in the Euclidean plane, except for the trivial case $k = 1$. The finite plane of \PMLinkEscape{order} $2$ is known as the \emph{Fano plane} and I will put a drawing of it sometime soon, when I figure out how to make diagrams. |
The above $k$ is the \emph{\PMLinkEscape{order}} of the finite plane. It is not known in general if finite planes exist of \PMLinkEscape{order} other than $k$ a power of a prime. The terminology "finite plane" is suggestive, as we can think of the edges as a finite collection of lines in Euclidean space in general position so that they all intersect pairwise in exactly one point. The added restriction that all pairs of vertices determine an edge (a line), however, makes it impossible to depict a finite plane in the Euclidean plane, except for the trivial case $k = 1$. The finite plane of \PMLinkEscape{order} $2$ is known as the \emph{Fano plane} and I will put a drawing of it sometime soon, when I figure out how to make diagrams. |
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