# 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:

- $(1): \quad b k = r v$

- $(2): \quad \lambda \paren {v - 1} = r \paren {k - 1}$

- $(3): \quad b \dbinom k 2 = \lambda \dbinom v 2$

- $(4): \quad k < v$

- $(5): \quad r > \lambda$

All of $v, b, r, k, \lambda$ are integers.

Some sources prefer to report $(3)$ as:

- $b = \dfrac {\dbinom v 2} {\dbinom k 2} \lambda$

which is less appealing visually, and typographically horrendous.

## Proof

- $(1)$: We have by definition that each point is in exactly $r$ blocks, and each block is of size $k$.

We have that $b k$ is the number of blocks times the size of each block.

We also have that $r v$ is the number of points times the number of blocks each point is in.

The two must clearly be equal.

- $(2)$: Comparing the left and right hand sides of the equation we can see that:

left hand side: An arbitrary point must be paired with $v-1$ other points.

If $\lambda>1$ then every point is paired $\lambda \paren {v - 1}$ times.

right hand side: An arbitrary point is paired with $k-1$ other points for each of the $r$ blocks it is in.

Therefore it is paired $r \paren {k - 1}$ times.

Both values give the number of times an arbitrary point is paired, therefore left hand side = right hand side.

- $(3)$: From equation $(1)$, we have that $r = \dfrac {b k} v$

From $(2)$ we have that:

- $r = \dfrac {v - 1} {k - 1} \lambda$

So:

\(\displaystyle \frac {b k} v\) | \(=\) | \(\displaystyle \frac {v - 1} {k - 1} \lambda\) | substituting for $r$ | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle b k \paren {k - 1}\) | \(=\) | \(\displaystyle \lambda v \paren {v - 1}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle b \binom k 2\) | \(=\) | \(\displaystyle \lambda \binom v 2\) | Binomial Coefficient with Two |

$\blacksquare$