The Erd(?)s-Ko-Rado Theorem For Affine Singular Linear Spaces

The Erd(?)s-Ko-Rado theorem is one of the most important results in the theory of extremal sets.The Erd(?)s-Ko-Rado theorem gives an upper bound on the cardinality of a intersecting family consisting of some 6)-subsets in a given 9)-set and describes exactly the structures of intersecting families which meet this bound.This theorem not only has a wide range of applications to association schemes,-designs and graphs,but also has natural extensions to vector spaces over finite fields,singular linear spaces,affine spaces,bilinear form graphs and so on.In this thesis,we give the upper bounds of the cardinalities of a 0-intersecting family and an -intersecting family( ? 1),respectively,for the affine singular linear space and the structures of the intersecting families which meet the upper bound by investigating the monotonicity of two certain functions and using the counting formulas of affine singular linear spaces.Thus,we prove the corresponding Erd(?)s-KoRado theorem for the affine singular linear spaces.