# balanced Incomplete Block Design, combinatorics

Give an example of BIBD(balanced Incomplete Block Design) with no repeated blocks in which λ > k.

Here which I need to find a BIBD where λ(each pair of vertices occurs together in exactly λ blocks) is greater than k(each block contains k vertices). For example, (v,b,r,1,2).

any suggestions?

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I'm sure you can find a bunch of 2-sets such that each pair of elements appears in exactly one of the 2-sets. Have you tried? – Gerry Myerson Apr 18 '12 at 10:55
Sorry, comment above was nonsense. But $\lambda=2$, $k=1$ can't possibly work - how can a pair of vertices appear in any blocks at all if each block has only one vertex? $\lambda=3$, $k=2$ can only be achieved if you allow multiple copies of some blocks. So I think you have to go to $\lambda=4$, $k=3$ to get any useful solution. See my answer. – Gerry Myerson Apr 19 '12 at 1:56

Take a set of size 6, and let the blocks be the subsets of size 3. I think you'll find this works with $\lambda=4$, $k=3$.
$r$ is the number of blocks containing a given vertex, and in the example I have suggested, this is not 3. How many 3-subsets of a given 6-set contain a given element of the 6-set? – Gerry Myerson Apr 19 '12 at 5:26
Yes, the number of blocks is the number of 3-subsets of a 6-set, which is 6-choose-3. I'm not sure I understand your matrix, but if what you are saying is one row for each vertex and one column for each block, and a 1 in position ij if vertex i is in block j, and a 0 in position ij otherwise, then the sum of the columns gives you $r$, the sum of the rows gives $k$, and you can use this to picture the problem. – Gerry Myerson Apr 19 '12 at 6:10