October 8 Math Colloquium

Dinesh Sarvate, CofC

A few results on t-group divisible designs (t-GDDs) for t=2 and 3

A combinatorial design is a collection of subsets (called blocks) of a given finite set. Depending on the conditions imposed on the collection, one gets different types of combinatorial designs.
A famous design is the Fano plane or a 2-design: 2-(7,3,1). Where the conditions imposed are blocks are subsets of a set of size 7 of same size 3 and any two distinct elements occur in exactly one block. The blocks of this design can be generated by adding every element of {0,1,2,3,4,5,6} in {0,1,3} modulo 7.

We start with a construction of a 2-GDD obtained with Dinkayehu Woldemariam of Ethiopia. Then we show how small 3-GDDs have been used in the construction of 3-designs, for example, by Hanani to prove that the necessary conditions are sufficient for the existence of 3-(v,4,λ). We will also see how my co-authors have used them to prove that necessary conditions are sufficient for the existence of several families of 3-GDDs with block size 4.

All required definitions will be given.

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