The full metamorphosis of λ-fold block designs with block size four into λ-fold triple systems

Abstract

Let (X, B) be a lambda-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 boolean OR S*) is a lambda-fold triple system, called a metamorphosis of the lambda-fold block design (X, B) into a lambda-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, T-i) is a partial lambda-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 boolean OR S-i*) is a lambda-fold triple system and we say that M-i = (X, T-i boolean OR S-i*) is the ith 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 a lambda-fold block design with block size 4 having a full metamorphosis into lambda-fold triple systems?