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A simple proof that all 1-designs exist

✍ Scribed by David Billington

Publisher: Elsevier Science
Year: 1982
Tongue: English
Weight: 68 KB
Volume: 42
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(82)90228-x

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✦ Synopsis

A short and elementary proof is given of the known result that a 1-(v,/~, r) design with b blocks exists if and only if vr = bk and b <~ (~).

A 1-(v, k, r) design is a set of b k-sets, called blocks, chosen from a v-set, say V, so that no two blocks are the same and each clement of V occurs exactly r times. It is easy to see that if a 1-(v, k, r) design exists with b blocks, then b ~ (~) and vr = bk.

The converse is apparently w~ll-known, and indeed it can be extracted from [1] and also from . However [1] involves several non-elementalT results, and the proof in [2] is not short.

We need to generalise the idea of a 1-design. A (k, r~ ..... r~)-dcsign is a set of b k-sets, called blocks, chosen from a v-set, say V :--{1, 2 ..... v}, so that no two blocks are the same, and for all i ~ V, the number of times i occurs is r~. Hence a 1-(v, k, r) design is a (k, r~ ..... r~)-design in which r~ = r for all i e V.

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