On blocking sets in a design
A 2-\((\nu,k, \lambda)\) design \(D\) \((2<k<\nu, \lambda>0)\) is a pair \((V,B)\), where \(V\) is a set with \(\nu\) elements and \(B\) is a collection of subsets (also called blocks) of \(V\), each subset with \(k\) elements, such that any two points are incident with exactly \(\lambda \) blocks. An intersection \(s\)-set in a 2-\((\nu,k, \lambda)\) design \(D\) is a set \(S\) with \(s\) elements of \(V\) such that each block intersects \(S\). A blocking \(s\)-set in a 2-\((\nu,k, \lambda) \) design \(D\) is an intersection \(s\)-set containing no block of \(D\). The author presents a bound for the cardinality of an intersection set of a 2-\((\nu,k, \lambda)\) design, and also investigates the existence of the blocking sets of type \((1,s)\) in a design. Some of the results of this paper are as follows: (I) Let \(S\) be an intersection \((s,1,n)\)-set of a 2-\((\nu,k, \lambda)\) design. Then \[ {nr+ \lambda- \sqrt {(nr+ \lambda)^2 -4\lambda nb} \over 2 \lambda} \leq s\leq {nr+ \lambda+ \sqrt {(nr+ \lambda)^2 -4\lambda nb} \over 2\lambda} \] where \[ b= {\lambda \nu (\nu-1) \over k(k-1)} \quad \text{and} \quad r= {\lambda (\nu-1) \over k-1}. \] Moreover, the equalities hold if and only if \(S\) is of type \((1,n)\). (II) If in a 2-\((\nu,k, \lambda)\) design \(D\) there exists a blocking \(s\)-set \(S\) of index 1, then \(\nu\leq (k-1)^2\). (III) In a 2-\(((k-1)^2, k, \lambda)\) design \(D\), with \(\lambda\leq k\), there is no blocking \(s\)-set of type \((1,s)\). (IV) If in a 2-\((Pk+ 1,k, \lambda)\) design \(D\) with \(2\leq P\leq k-3\) there is a blocking \(s\)-set \(S\) of type \((1,s)\), then it holds that \(P\leq \lambda-1\) and \(k\leq \lambda^2-\lambda+1\).
scientific article; zbMATH DE number 1110118
scientific article