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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:
(1)
$${\mathcal G}$$ is a partition of $$X$$ into subsets called groups;
(2)
$${\mathcal B}$$ is a collection of subsets of $$X$$, called blocks, such that a group and a block contain at most one element in common; and
(3)
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.

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