Some results on (v,{5,w*})‐pbds

In this article, we construct pairwise balanced designs (PBDs) on v points having blocks of size five, except for one block of size w ϵ {17,21,25,29,33}. A necessary condition for the existence of such a PBD is v ⩾ 4 w + 1 and (1) v ≡ 1 or 5 (mod 20) for w = 21, 25; (2) v ≡ 9 or 17 (mod 20) for w = 17,29; (3) v ≡ 13 (mod 20) for w = 33. We show that these necessary conditions are sufficient with at most 25 possible exceptions of ( v,w ). We also show that a BIBD B (5, 1; w ) can be embedded in some B (5, 1; v ) whenever v ≡ w ≡ 1 or 5 (mod 20) and v ⩾ 5 w − 4, except possibly for ( v, w ) = (425, 65). © 1995 John Wiley & Sons, Inc.