ABSTRACT
BIBD(4,4,2) can be obtained by taking the block {1,2,3,4} twice. (It is degenerate because v = k, but it is sufficient here.) A BIBD(7,4,2) is obtained by developing the set {0, 1, 2, 4}modulo 7. Now gcd(7 ·6, 4 ·3) = 6, so β(S)|6. Also {4, 7} ⊂ S. Therefore, S is eventually periodic, with period 3; so for sufficiently large v ≡ 1 (mod 3), there exists a BIBD(v, 4, 2), simply by the closure theorem. However, B({4, 7}) = {v : v ≡ 1 (mod 3), v 6= 10, 19}, so there exists a BIBD(v, 4, 2) for all positive v ≡ 1 (mod 3) with the possible exception of v = 10 or v = 19. (It is for this reason that the study of closures is important.) Because there exists a BIBD(v, 4, 2) for v = 10 and v = 19, S = {v : v ≡ 1 (mod 3)}.
