The set of values of \(b\) for which there exist block designs with \(b\) blocks (Q1772423)

This paper on block designs is intended to provide the reader with motivation and background material while detailed proofs are presented in the electronic version of Discrete Mathematics. Let \(B\) denote the set of those values of \(b\) for which a block design with \(b\) blocks exists. Let \(B_k= \{b\mid\) there exists a \((\nu,b,r,k,\lambda)\) design for some \(\nu\), \(r, k\geq 3\), and with \(\lambda= 1\}\). This paper contains some results providing information about the sets \(B_k\) and \(B\). The discussion is primarily concerned with addressing the following problem: For certain integers \(h\), do there exist integers \(k\) and \(k'\) such that \(b'= b+ h\) has infinitely many solutions \(b\), \(b'\) which satisfy \(b'\in B_k'\)? A number of results concerning the values of \(k\), \(k'\) and \(h\) (\(k\geq 3\), \(k'\geq 3\), \(k\neq k'\)) for which \(B(k,k',h)\) is empty are presented where \(B(k,k',h)= \{b\mid b\in B_k\) and \(b+h\in B_k'\}\).