块大小为3、4和5的五边形几何图形
Pentagonal geometries with block sizes 3, 4 and 5
摘要：
五边形几何PENT（$k$，$r$）是一个部分线性空间，其中每条线或块都与$k$点关联，每个点都与$r$线关联，并且对于每个点$x$，都有一条线恰好与那些不与$x$共线的点关联。五边形几何中的相对线对由两条平行线组成，使得其中一条线上的每个点与另一条线上的那些点不精确地共线。给出了块大小为3的连通亏图的五角几何无穷序列的直接构造。此外，我们提出了39个新的五边形几何块大小为4和5块大小为5，所有连接的亏损图。因此，我们确定存在谱，直到不包含相反线对的PENT（4，$r$）和具有一个相反线对的PENT（4，$r$）的几个可能的例外。更一般地说，给定$j$，我们表明，存在一个PENT（4，$r$）与$j$相反的线对所有足够大的容许$r$。利用一些新的区组大小为5的可分组设计（包括$2^{35}$，$2^{71}$和$10^{23}$），我们显著地扩展了PENT（5，$r$）的存在谱.
英文摘要：
A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4, $r$) that do not contain opposite line pairs and for PENT(4, $r$) with one opposite line pair. More generally, given $j$ we show that there exists a PENT(4, $r$) with $j$ opposite line pairs for all sufficiently large admissible $r$. Using some new group divisible designs with block size 5 (including types $2^{35}$, $2^{71}$ and $10^{23}$) we significantly extend the known existence spectrum for PENT(5, $r$).
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