Generalizations of results of Stanton on BIBDs (Q1193712)

Characterization theorems using intersection numbers for designs made up by putting together \(n\) copies of a symmetric design are proved. For instance it is proved that if a design \(D\) on \(v\) points with block size \(k\) has the property that every block has \(n-1\) intersection numbers equal to \(k\) and the other \(b-n\) intersection numbers are equal, then \(D\) is made up of \(n\) copies of a symmetric design. This generalizes the result by \textit{R. G. Stanton} for \(n=2\) [Sankhyā, Ser. A 32, 457-458 (1970; Zbl 0234.62037)]. On the other hand if \(D\) has one block having \(n-1\) intersection numbers equal to \(k\) and the other \(b-n\) numbers being equal, then \(D\) has parameters \(b=nv\), \(r=nk\) and \(\lambda=n\lambda_ 0\) with \(\lambda_ 0(v-1)= k(k-1)\). Another set of theorems in this papers are embedding theorems. For instance it is proved that any \(\text{BIBD}(v,3,\lambda)\) can be embedded into a \(\text{BIBD}((\lambda+2)v,3,\lambda)\) which generalizes a result of \textit{R. G. Stanton} and \textit{I. P. Goulden} for \(\lambda=1\) [Aequationes Math. 22, 1-28 (1981; Zbl 0466.05011)].