Let us see some examples. Trivial designs are single block consisting of all points or all -subsets of . These are often not focus of study in designs.
As a first non-trivial example, a symmetric Steiner system can be easily represented as triangle on points with the lines denoting the blocks.
This is simply the symmetric Steiner system . More geometrically,
Alternatively, set and .
More generally, symmetric Steiner systems are called as projective planes and we shall see more about them later. Why implies that follows from the below exercise which is via double counting of pairs of -sets and blocks containing them.
Show that in a design
Let and . Clearly . Take . Then clearly there exists a unique such that . Thus, any pair of points belongs to exactly one block i.e., this is a .
A second Steiner system on is as follows : . This is a . If we take blocks containing and delete , we get , the previous Steiner system.
Take the edges of as points. Let the blocks be sets of edges that either are the edges of a perfect matching or the edges of a triangle. Show that this is . Is it isomorphic to the design in the above example ?
One of the first properties of designs is regularity as sets i.e., every point belongs to the same number of sets, say . is called as replication number.
Let be a design containing blocks. Then every point is contained in exactly blocks where satisfies
We will prove by double counting. Let and suppose its replication number is . Let us count the cardinality of the set .
For the points , there are exactly blocks containing both . This gives the cardinality to be
On the other hand, there are blocks containing and each of the blocks contains other points. So the cardinality Thus we have . This gives that is independent of . Now the second identity follows by double counting the set . ∎
In other words, the above shows that a design is also a design. More generally, we have the following.
Let be a design and let . Then it is also a design for
Fix an -set . Count the number of pairs such that is a -set containing and . Summing over gives that and summing over gives that . ∎
Let and is a -subset of . The number of blocks of an design that contain none of the points of is
Let be a design with blocks and replication number . Prove that the complement is a design if . Determine the parameters.