The Complete Nontrivial-Intersection Theorem for Systems of Finite Sets

The theorem presented and proved in this paper can be viewed as an extension or improvement of our recent Complete Intersection Theorem [1] and may be called the Complete Nontrivial-Intersection Theorem. It goes considerably beyond the well-known Hilton Milner Theorem [10]. We put the result into the proper perspective with a brief sketch of the key steps in its development, beginning with the pioneering paper [4] by Erdo s, Ko, and Rado. Since we again use the methods from [1], we also keep the notation from this earlier paper as far as possible. N denotes the set of positive integers and for i, j # N, i< j, the set [i, i+1, ..., j] is abbreviated as [i, j]. For k, n # N, k n, we set