On the 3BD-closed set B3({4,5,6})

Let B3(K) = {v:∃ an S(3,K,v)}. For K = {4} or {4,6}, B3(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if v ∈ B3 ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B3{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ⊂ B3({4,5,6}) by Hanani and that B3({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ⊂ B3({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B3({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.