Projective \((2n,n,\lambda,1)\)-designs (Q1170244)

The paper deals exclusively of \(\lambda\)-covers, i.e. of sets \(S\) with \(2n\) elements with a system of blocks of \(n\) elements such that each point of \(S\) is in \(\lambda\) blocks and every two blocks have a non-empty intersection. The problem of existence of such covers with given parameters is completely solved in the paper. Interesting results on the existence of subcovers and on extensions of a cover with a not too great \(\lambda\) to a \((\lambda+2)\)-cover on the same set are obtained. As conclusion, a set of open problems with some remarks is given. Proofs are mainly by construction, by induction, by cases and by quotations, using graph theory too.