The minimum independence number for designs
Let \(V\) be a set with \(n\) elements, and let \(D\) denote a collection of subsets of \(V\), each subset with \(k\) elements in it, such that every subset of \(t\) elements of \(V\) appears exactly in \(\lambda\) blocks of \(D\). Such a configuration is called \(t\)-\((n, k, \lambda)\) design and is denoted by a pair \(P= (V, D)\). If no block of \(D\) is contained in a subset \(S\) of \(V\), then \(S\) is called independent in \(P\). The maximum cardinality of any independent set in \(P\) is called the independence number and is denoted by \(\alpha(P)\). This paper deals with the problem of obtaining an upper bound on the minimum independence number of all designs of order \(n\). The main result is stated as follows: Consider designs with \(t= 2, 3\) and let \(k\geq 2t- 1\). There exists a constant \(c\) such that if \(q\) is a sufficiently large prime power and if there exists a \(t\)-\((q+ t- 2, k, \lambda)\) design then there exists a \(t\)-\((n, k, \lambda)\) design \(P\) with \(n= q^ 2+ t- 2\) such that \(\alpha(P)\leq c(n)^{{k- t\over k- 1}}(\ln n)^{{1\over k- 1}}\), where \(c\) is not greater than a certain number \(N(k, t, \lambda)\).
scientific article; zbMATH DE number 778332
scientific article