Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 $\lambda=4$, $k=3$.

share|cite|improve this answer
then which means that v=6, r=3, k=3, λ=4.correct? then using the formula, r(k-1)=λ(v-1), 3(2)=4(5) which does not equal. Do I missing something? – user28699 Apr 19 '12 at 4:52
$r$ 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
yeah, I guess that is my problem.... it's hard to picture them just by looking at the figures. It sometimes help me picturing of matrix where rows are the vertices and columns are the blocks, and sum of each rows give us the r-regular and sum of columns gives us k-uniform, is it correct and can I apply this picturing in this problem? – user28699 Apr 19 '12 at 6:01
so there will be 6 choose 3 many different blocks, along with vertices 1 through 6, correct? – user28699 Apr 19 '12 at 6:06
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.