7:12 PM
Converting our problem into a BIBD:
the set of varieties $X = {v_1, v_2, \ldots, v_{32}}$ are the red nodes
the blocks in $\mathcal B = \{\!\{b_1, b_2, \ldots, b_{28}\}\!\}$ are the adjacencies of the blues nodes 1 through 28
im using that notation for multiset but in this case I think it's a set
so $b_3$ are all the varieties adjacent to red node $v_3$
not sure how many times $r$ each variety appears across all adjacencies, leave it undetermined
each blue vertex allows for 8 adjacencies, so $k=8$
if I'm correct that the 2-multiplicity of length-2 paths between red nodes corresponds to each pair $v_i, v_j: i \ne j$ occurring in exactly 2 blocks, then $\lambda = 2$
by the second equation of block designs, $r(k-1) = \lambda(v-1)$
so $r \cdot 7 = 2 \cdot 31$ which doesn't have an integer solution
so the naive approach, of taking all parameters to be maximal according to the physical constraints, does not lend itself to a BIBD
and actually the parameters are rigid, since $\lambda \cdot (v-1) = 2 \times 31$ is irreducible
or whatever the right word is for can't-be-factored-further
so we drop the condition that the design be balanced
a pairwise balanced design PBD is like a BIBD but $k$ need not be constant
or, actually, better idea
seek a BIBD for $v < 32$, and then the $32-v$ left will be dealt with non-optimally
I guess the first one to try would be $v = 29$
