Existence of indecomposable and simple block design \(B(5,4;v)\) (Q1310518)

A 2-\((v,k,\lambda)\) design is said to be simple resp. indecomposable if it has no repeated blocks resp. if no subset of the set of blocks yields a 2-\((v,k,\lambda_ 1)\) design for some \(\lambda_ 1< \lambda\). It is the main purpose of the present paper to determine the spectrum \(K\) of all \(v\)'s for which a simple indecomposable 2-\((v,5,4)\) design exists. The authors prove namely that the necessary conditions \(\lambda(v- 1)\equiv 0\pmod{k-1}\), \(\lambda v(v- 1)\equiv 0\pmod{k(k-1)}\), \(\lambda\leq\binom{v-2}{k-2}\) for the existence are also sufficient in this case. In other words the required design exists if and only if \(v\equiv 0\) or \(1\pmod{5}\), \(v\geq 6\). The corresponding result with repeated blocks allowed and no request for indecomposability was proved in \textit{H. Hanani} [The existence and construction of balanced incomplete block designs, Ann. Math. Stat. 32, 361-386 (1961; Zbl 0107.361)] and again in \textit{H. Hanani} [On balanced incomplete block designs with blocks having five elements, J. Comb. Theory, Ser. A 12, 184-201 (1972; Zbl 0247.05009)]. The authors prove first of all that \(K\) is PBD-closed (i.e. if \(H\) denotes the set of all \(n\)'s for which a 2-design of \(n\) points with \(\lambda= 1\) and block sizes in \(K\) exists, then \(H= K\)). This fact and some recursive techniques based on group divisible designs allow the authors to reduce their main result to show the existence of the required design when \(v\) is one of 58 given values between 10 and 181. Of course settling existence for such values requires some `ad hoc' work, which occupies the most part of the paper.