On the 3BD‐closed set B 3 ({4,5,6})

Let B 3 ( K ) = { v :∃ an S (3, K , v )}. For K  = {4} or {4,6}, B 3 ( 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  ∈  B 3 ({4, 5, 6}), then v  ≡ 0, 1, 2 (mod 4). It is known that B 3 {4, 6}) = { v  > 0: v  ≡ 0 (mod 2)} ⊂ B 3 ({4,5,6}) by Hanani and that B 3 ({4, 5}) = { v  > 0: v  ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v  ≠ 13} ⊂ B 3 ({4, 5, 6}). We shall focus on the case of v  ≡ 9 (mod 12). It is proved that B 3 ({4,5,6}) = { v  > 0: v  ≡ 0, 1, 2 (mod 4) and v  ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.