This homework is about a highly constrained hypergraph, called a 2-design. Ronal

Question

This homework is about a highly constrained hypergraph, called a 2-design. Ronald Fisher, a population geneticist, was interested in studying n varieties of plants under m different growing conditions, each of which he called a block. He wanted · each block to grow k n different varieties of plants (and no duplicates). ·any single variety to grow in exactly r blocks, . and any pair of distinct varieties to grow together in exactly blocks. Ronald Fisher wanted to find out how few1 blocks he could use create a 2-design for n varieties of plants. The challenge problem shows m 2 n, that is, he always needs more blocks than varieties of plants. It is not even known for which n we can achieve2 with m n and -1.

Explanation / Answer

As per the given criteria:-

m = number of hyperedegs
= 7
n = number of vertices
= 7
It is a k-uniform hypergraph where k=3,
since each hypheredge is containing 3 nodes.
It is a r-regular hypergraph where r=3,
since each vertex is lies in exactly 3 edges.
Yes, it is a 2-design hypergraph because it is a lambda-regular uniform hypergraph where lambda =3

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