A generalization of a theorem of Dehon for simple \(t\)-designs (Q1924177)

The main result of the paper is the following theorem that generalizes a construction by \textit{M. Dehon} [Discrete Math. 90, No. 2, 137--142 (1991; Zbl 0746.05010)]: Let \({\mathcal D}=({\mathcal X},{\mathcal B})\) be a \(t\)-\((v,w,\lambda)\) design, and let \({\mathcal D}'=({\mathcal X}',{\mathcal B}')\) be a simple \(t\)-\((v,k,\lambda')\) design with \(|B'\cap C'|<k-h\) for any two distinct blocks \(B',C'\in{\mathcal B}'\), \(0\leq h\leq h\). If \[ \lambda^{\prime 2}_0(\lambda- 1)\begin{pmatrix} k\\ h\end{pmatrix} \begin{pmatrix} k\\ t\end{pmatrix} \sum^h_{i=0} {{k-i\choose h-i}\over {k-i\choose t}}<\begin{pmatrix} w\\ k-h\end{pmatrix}, \] then there exists a simple \(t\)-\((v,k,\lambda\lambda')\) design in which any two blocks meet in less than \(k-h\) points.