I have to determine whether or not the list of sets with the parameters v=4, b=6, r=3, k=2 and $\lambda$ = 1 is a valid block design. Now my first thought was to ensure that these parameters satisfy the conditions:
- b $\geq$ v
- bk = vr
- $\lambda$ (v-1) = r (k-1).
And they did satisfy the conditions. However this is not sufficient according to my lecturer to show its a valid block design. So my next step was to identify the design as an affine plane of order n=2 and since an affine plane is a symmetry block design then the list of sets do form a valid block design. However, I am not sure if this is enough either. Can someone help me out please?
It is not sufficient to check that these parameters satisfy the equations you've listed. These equations are necessary conditions: if the parameters violate them, then there can be no block design. But they do not guarantee that there do exist block designs with a set of parameters.
It is sufficient to invoke the affine plane of order two. To be precise, what you are saying here is "Yes, a block design with these parameters does exist. The affine plane of order $2$ is an example."
The best answer would often be to say this, and also list what the points and blocks of the affine plane of order $2$ are. These parameters are very small, and it can be easily checked by hand that the set system with $v=4$ points $\{1, 2, 3, 4\}$ and $b=6$ blocks $$\{1,2\}, \{1,3\}, \{1,4\}, \{2,3\}, \{2,4\}, \{3,4\}$$ is a block design with the parameters we wanted. Additionally, you don't have to think of this block design as an affine plane; it is also the block design obtained by taking all $2$-element subsets of the set of points.
(That's a less interesting but also valid family of block designs, giving a $b = \binom v2$, $r=v-1$, $k=2$, $\lambda=1$ block design for any $v$.)