Room Squares Generalized

Publisher This chapter discusses the generalizations of room squares. The emphasis is on existence and enumeration or embedding. A feature that is common to all the structures considered in the chapter is that they consist of cells that are either empty or contain subsets of a basic n-set N. The set of subsets occurring in the nonempty cells forms the set of blocks of a combinatorial design called the “underlying design” of the structure. A room square of order n (RS(n)) is a square array such that (1) every cell of the array is either empty or contains a 2-subset of an n-set N, (2) every element of N is contained in exactly one cell of each row (column), and (3) every 2-subset of N is contained in exactly one cell of the array. The chapter also discusses recursive constructions for RS(n) that include Moore-type constructions and doubling, as well as other multiplication constructions.