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The asymptotic existence of resolvable group divisible designs. (English) Zbl 1262.05016
Summary: A group divisible design (GDD) is a triple \((X,\mathcal{G},\mathcal{B})\) which satisfies the following properties:
\({\mathcal G}\) is a partition of \(X\) into subsets called groups;
\({\mathcal B}\) is a collection of subsets of \(X\), called blocks, such that a group and a block contain at most one element in common; and
every pair of elements from distinct groups occurs in a constant number \(\lambda\) blocks.
This parameter \(\lambda\) is usually called the index. A \(k\)-GDD of type \(g^u\) is a GDD with block size \(k\), index \(\lambda= 1\), and \(u\) groups of size \(g\). A GDD is resolvable if the blocks can be partitioned into classes such that each point occurs in precisely one block of each class. We denote such a design as an RGDD. For fixed integers \(g\geq 1\) and \(k\geq 2\), we show that the necessary conditions for the existence of a \(k\)-RGDD of type \(g^u\) are sufficient for all \(u\geq u_0(g, k)\).
As a corollary of this result and the existence of large resolvable graph decompositions, we establish the asymptotic existence of resolvable graph GDDs, \(G\)-RGDDs, whenever the necessary conditions for the existence of \((v,G,1)\)-RGDs are met. We also show that, with a few easy modifications, the techniques extend to general index.

05B30 Other designs, configurations
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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