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. $$b=\frac\lambda$$.

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

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

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