Intersection theorems for group divisible difference sets

New semiregular divisible difference sets

We modify and generalize the construction by McFarland (1973) in two different ways to construct new semiregular divisible difference sets (DDSs) with 21 ~ 0. The parameters of the DDS fall into a family of parameters found in Jungnickel (1982), where his construction is for divisible designs. The final section uses the idea of a K-matrix to find DDSs with a nonelementary abelian forbidden subg...

Intersection Theorems for Systems of Sets (iii)

A system or family (A. : y E N) of sets A y , indexed by the elements of a set N, is called an (a, b)-system if I N I = a and I A,, I = b for y c-N. Expressions such as "(a, < b)-system" are self-explanatory. The system (A,, : y c N) is called a A-system [1] if A,, n A,, = Ap n A Q whenever p, y, p, c c N ; p 7~ y ; p =A a. If we want to indicate the cardinality I NJ of the index set N then we ...

Intersection Theorems for Systems of Sets

A version of Dirichlet's box argument asserts that given a positive integer a and any a2 +1 objects x0 , x1 , . . ., xa 2, there are always a+1 distinct indices v (0 < v < a 2) such that the corresponding ad-1 objects x,, are either all equal to each other or mutually different from each other . This proposition can be restated as follows . Let N be an index set of more than a 2 elements and le...

Intersection Theorems for Systems of Sets ( Ii )

In this paper we present the complete solution of the problem which was considered in [1], with the exception of the case in which both the given cardinal numbers are finite. The results of [1] will not be assumed. We begin by introducing some definitions, f A system Li = (Bv: veiV) of sets Bv, where v ranges over the index set N, is said to contain the system Eo = (A^ : n e M) if, for (j.o e M...

Some nonexistence results on divisible difference sets