在1961年, Erdos, Ko和Rado给出了n元集合的k元子集构成的相交族的基数的最大值, 并证明基数达到最大值的相交族是平凡的. 在1986年, Hilton和Milner给出非平凡的相交k元子集族的基数的最大值, 并表明基数达到最大值的非平凡相交族的覆盖数为2.
 
相交族以及覆盖数的概念已被推广到有限域上的空间上. Hsieh给出了n维向量空间的相交族的基数的最大值. Ou, Lv和Wang确定了奇异线性空间的相交族的基数的最大值. Guo和Xu则给出了仿射空间的相交族的基数的最大值. 他们的结果表明基数达到最大值的相交族是平凡的. 在2013年, Blokhuis等人给出了向量空间的非平凡相交族的基数的最大值, 并且证明了达到最大值的集族的覆盖数为2.
 
本文研究三类空间的非平凡相交族, 并刻画达到最大值的集族的结构. 首先, 我们研究奇异线性空间的非平凡相交族的性质, 确定基数达到最大值的非平凡相交族的结构, 从而给出奇异线性空间的Hilton-Milner定理. 其次, 我们研究仿射空间的非平凡相交族的性质, 取定基数达到最大值的非平凡相交族的结构, 从而给出仿射空间的Hilton-Milner定理. 最后, 我们研究向量空间以及奇异线性空间的覆盖数达到最大的相交族的性质, 给出集族基数的最大值, 并刻画达到最大值的集族的结构. 

In 1961, Erdos, Ko and Rado [13] determined the maximum size of an intersecting family consisting of k-subsets of an n-set, and showed that any family with maximum size must be trivial. In 1986, Hilton and Milner [41] determined the maximum size of a non-trivial intersecting family, and showed that any non-trivial intersecting family with maximum size must have covering number 2. The concepts of intersecing family and covering number are extended to finite spaces over finite fields. Hsieh [42] determined the maximum size of an intersecting family of finite vector space. Ou, Lv and Wang [56] determined the maximum size of an intersecting family of singular linear space. Guo and Xu [38] determined the maximum size of an intersecting family of affine space. Their results showed that families with maximum size must be trivial. In 2013, Blokhuis et al. determined the maximum size of an non-trivial intersecting family of vector space, and showed that the covering number of a non-trivial intersecting family with maximum size is 2. This thesis studies non-trivial intersecting families based on three types of spaces, and determines the structures of non-trivial intersecting families with maximum size. First, we determine the maximum size of non-trivial intersecting families based on singular linear space, and describe the structure of intersecting family which reaches the maximum size. Thus, the Hilton-Milner theorem based on singular linear space has been proved. Next, we determine the maximum size of non-trivial intersecting families based on affine space, and describe the structure of intersecting family which reaches the maximum size. Thus, the Hilton-Milner theorem based on affine space has been proved. At last, we determine the maximum size of an intersecting family of vector space with maximum covering number, and describe the structure of intersecting family which reaches the maximum size.