Necessary Condition for Existence of BIBD

Theorem
Let there exist be a BIBD with parameters $v,b,r,k,\lambda$. Then the following are true:

1. $bk=rv$,

2. $\lambda(v-1)=r(k-1)$,

3. $\displaystyle b=\frac{ {v \choose 2} }{ {k \choose 2} }\lambda$.

NOTE: The above are all integers.

Proof
1. Since each point is in exactly $r$ blocks, and each block is of size $k$. Then $bk$(the number of blocks times the size of each block) must be equal to $rv$(the number of points times the number of blocks each point is in).

2. Comparing the left and right hand sides of the equation we can see that:

LHS: An arbitrary point must be paired with $v-1$ other points. If $\lambda>1$ then every point is paired $\lambda(v-1)$ times.

RHS:An arbitrary point is paired with $k-1$ other points for each of the $r$ blocks it is in. Therefore it is paired $r(k-1)$ times.

Both values give the number of times an arbitrary point is paired, therefore LHS=RHS

3. From equation 1, we have that $\displaystyle r=\frac{bk}{v}$, and from 2 we have that $ r=\frac{v-1}{k-1}\lambda$.

Substituting $r$ we get that $\displaystyle \frac{bk}{v}=\frac{v-1}{k-1}\lambda$.

$\displaystyle \implies b=\frac{v(v-1)}{k(k-1)}\lambda=\frac{ {v \choose 2} }{ {k \choose 2} }\lambda$.