Let's say I have a set of elements $V$. I can use all subsets of $V$ of size $k$ to satisfy intersection conditions.
Example: $V = \{ 1, 2, 3, 4, 5\}$, $k = 3$. $|B_1 \cap B_2|=2$,$|B_2 \cap B_3|=2$,$|B_3 \cap B_1|=1$
One solution would be $\{ \{1, 2, 3\}, \{1, 2, 4\}, \{2, 4, 5\} \}$.
I'm wondering how to find/enumerate these solutions. The intersection conditions are limited if $B_4$ is added to the conditions.
Example: $V = \{ 1, 2, 3, 4, 5\}$, $k = 3$. $|B_1 \cap B_2|=2$,$|B_2 \cap B_3|=2$,$|B_3 \cap B_1|=\bf{2}$
One solution would be $\{ \{1, 2, 3\}, \{1, 2, 4\}, \{1, 2, 5\} \}$. If $B_4$ is added, then $|B_4 \cap B_i| = 2$ cannot be true for all $i$.
It would be also nice to know if I could find the possible values of $|B_4 \cap B_i|$ too. It looks like these values really depend on the contents of the first $B_i$ sets, although I'm not sure.
Associate each subset $B \subseteq \{1,2,3,4,5\}$ with a vector $v_{B} \in \mathbb{R}^{5}$ with 0/1 entries, such that the $j$th entry of $v_{B}$ is 1 if and only if $j \in B$. This is called the characteristic vector of $B$.
Now you have that $|B_{i} \cap B_{j}| = v_{B_{i}}\cdot v_{B_{j}}$.
So you are looking for vectors with 0/1 entries satisfying $v_{i}\cdot v_{i} = 3$ for all $1\leq i \leq 3$, $v_{1}\cdot v_{2} = 2$, $v_{2}\cdot v_{3} = 2$, $v_{1}\cdot v_{3} = 1$. This is equivalent to finding a $5\times 3$ 0/1 matrix $$A = \begin{bmatrix} v_{1} & v_{2} & v_{3} \end{bmatrix}$$ such that $$A^{T}A = \begin{bmatrix} 3 & 2 & 1\\ 2 & 3 & 2\\ 1 & 2 & 3 \end{bmatrix}$$
In general it is not easy to find all 0/1 solutions to such an equation, or even to determine if any 0/1 solution exists. You can assign variables to the entries of $A$ and get a system of homogeneous quadratic equations in 15 variables, there is special software that can solve this for you but it will take a lot of time if there are more than a few variables involved.