Existence of (v,{5,w∗},1)-PBDs

In this paper, 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. The necessary conditions for the existence of a (v,{5,w∗},1)-PBD with a distinguished block of size w are that v⩾4w+1, v≡w≡1 (mod 4) and either v≡w (mod 20) or v+w≡6 (mod 20). Previously, w⩽33 has been studied, and the necessary conditions are known to be sufficient for w=1, 5, 13 and 21, with 8 possible exceptions when w⩽33. In this article, we eliminate 3 of these possible exceptions, showing sufficiency for w=25 and 33. Our main objective is the study of 37⩽w⩽97, where we establish sufficiency for w=73, 81, 85 and 93, with 67 possible exceptions with 37⩽w⩽97. For w≡13 (mod 20), we show that the necessary existence conditions are sufficient except possibly for w=53,133,293 and 453. For w≡1,5 (mod 20), we show the necessary existence conditions are sufficient for w⩾1281,1505, and for w≡9,17 (mod 20), we show that w⩾2029,2477 is sufficient with one possible exceptional series, namely v=4w+9 when w≡17 (mod 20). We know of no example where v=4w+9. In this article, we also study the 4-RBIBD embedding problem for small subdesigns (up to 52 points) and update some results of Bennett et al. on PBDs containing a 5-line. As an application of our results for w=33 and 97, we establish the smallest number of blocks in a pair covering design with k=5 when v≡1 (mod 4) with 37 open cases, the largest being for v=489; hitherto, there were 104 open cases, the largest being v=2249.