The Full Metamorphosis of \(\lambda\)-Fold Block Designs with Block Size Four into \(\lambda\)-Fold Triple Systems

Abstract

Let \((X,{B})\) be an \(\alpha\)-fold block design with block size \(4\). If a star is removed from each block of \({B}\), the resulting collection of triangles \({T}\) is a partial \(\lambda\)-fold triple system \((X,{T})\). If the edges belonging to the deleted stars can be arranged into a collection of triangles \({S}^*\), then \((X,{T} \cup {S}^*)\) is an \(\lambda\)-fold triple system, called a metamorphosis of the \(\lambda\)-fold block design \((X, {B})\) into a \(4\)-fold triple system.

Label the elements of each block \(b\) with \(b_1, b_2, b_3\) and \(b_4\) (in any manner). For each \(i = 1,2,3,4\), define a set of triangles \({T}_i\) and a set of stars \({S}_i\) as follows: for each block \(b = (b_1, b_2, b_3, b_4)\) belonging to \({B}\), partition \(b\) into a triangle and a star centered at \(b_i\), and place the triangle in \({T}_i\) and the star in \({S}_i\). Then \((X,\mathcal{T}_i)\) is a partial \(\alpha\)-fold triple system.

Now if the edges belonging to the stars in \({S}_i\) can be arranged into a collection of triangles \({S}_i^*\), then \((X,{T}_i \cup {S}_i^*)\) is an \(\lambda\)-fold triple system and we say that \(M_i = (X,{T}_i \cup {S}_i^*)\) is the \(i\)th metamorphosis of \((X,{B})\).

The full metamorphosis of \((X,{B})\) is the set of four metamorphoses \(\{M_1, M_2, M_3, M_4\}\). The purpose of this work is to give a complete solution of the following problem: For which \(n\) and \(\lambda\) does there exist an \(\lambda\)-fold block design with block size \(4\) having a full metamorphosis into \(\lambda\)-fold triple systems?