Further results on -PBDs

In this article we investigate the existence of pairwise balanced designs on v points having blocks of size five, with a distinguished block of size w , briefly ( v , { 5 , w ∗ } , 1 ) -PBDs. cessary conditions for the existence of a ( v , { 5 , w ∗ } , 1 ) -PBD with a distinguished block of size w with v > w are that v ≥ 4 w + 1 , v ≡ w ≡ 1 ( mod 4 ) and either v ≡ w ( mod 20 ) or v + w ≡ 6 ( mod 20 ) . Previously, Bennett et al. had shown that these conditions are sufficient for w > 2457 with the possible exception of v = 4 w + 9 when w ≡ 17 ( mod 20 ) , and had studied w ≤ 97 in detail, showing there that the necessary conditions are sufficient with 71 possible exceptions. s article, we show sufficiency for w ≡ 1 , 5 , 13 ( mod 20 ) and give a small list of possible exceptions containing 26 and 104 values for w ≡ 9 , 17 ( mod 20 ) . For w ≡ 9 ( mod 20 ) , all possible exceptions satisfy either v = 4 w + 13 with w ≤ 489 or v ≢ w ( mod 20 ) with v < 5 w and w ≤ 129 ; for w ≡ 17 ( mod 20 ) , all possible exceptions except ( v , w ) = ( 197 , 37 ) , ( 529 , 37 ) satisfy either v = 4 w + 9 with w ≤ 1757 or v ≢ w ( mod 20 ) with v < 5 w and w ≤ 257 . application of our results for w = 97 , we establish that, if v ≡ 9 , 17 ( mod 20 ) , v ≥ 389 and v ≠ 429 , then the smallest number of blocks in a pair covering design with k = 5 is ⌈ v ( v − 1 ) / 20 ⌉ , i.e., the Schönheim bound.