论文标题
Erdős-ko-Rado定理的稳定性在圆几何形状中
Stability of Erdős-Ko-Rado Theorems in Circle Geometries
论文作者
论文摘要
圆几何形状是捕获3维空间中球体，锥和倍曲底圆圈几何形状的发射率结构。在上一篇论文中，作者表征了有限卵形圆几何形状中最大的相交家族，除了奇数秩序的莫比乌斯平面。在本文中，我们表明，在这些莫比乌斯平面中，如果订单大于3，则最大的相交家族是通过固定点的圆圈集。我们在唯一已知的有限非卵形圆几何体系列中显示出相同的结果。使用相同的技术，我们在所有卵形圆几何形状中显示了大型相交家族的稳定性。更具体地说，我们证明了一个已知的有限圆圈$ q $的有限圆几何形状中的一个相交的家庭$ \ Mathcal f $，带有$ | \ Mathcal f | \ geq \ frac 1 {\ sqrt2} q^2 + 2 \ 2 \ sqrt 2 q + 8 $，必须由圆圈组成，或者在均匀顺序的laguerre平面的情况下通过公共核组成。
Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite non-ovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family $\mathcal F$ in one of the known finite circle geometries of order $q$, with $|\mathcal F| \geq \frac 1 {\sqrt2} q^2 + 2 \sqrt 2 q + 8$, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.
