Existence of <f>(v,{5,w&ast;},1)</f>-PBDs

In this paper, we investigate the existence of pairwise balanced designs on <f>v</f> points having blocks of size five, with a distinguished block of size <f>w</f>, briefly <f>(v,{5,w&ast;},1)</f>-PBDs. The necessary conditions for the existence of a <f>(v,{5,w&ast;},1)</f>-PBD with a distinguished block of size <f>w</f> are that <f>v&ges;4w+1</f>, <f>v≡w≡1 (mod 4)</f> and either <f>v≡w (mod 20)</f> or <f>v+w≡6 (mod 20)</f>. Previously, <f>w&les;33</f> has been studied, and the necessary conditions are known to be sufficient for <f>w=1</f>, <f>5</f>, <f>13</f> and <f>21</f>, with <f>8</f> possible exceptions when <f>w&les;33</f>. In this article, we eliminate <f>3</f> of these possible exceptions, showing sufficiency for <f>w=25</f> and 33. Our main objective is the study of <f>37&les;w&les;97</f>, where we establish sufficiency for <f>w=73</f>, 81, 85 and 93, with 67 possible exceptions with <f>37&les;w&les;97</f>. For <f>w≡13 (mod 20)</f>, we show that the necessary existence conditions are sufficient except possibly for <f>w=53,133,293</f> and 453. For <f>w≡1,5 (mod 20)</f>, we show the necessary existence conditions are sufficient for <f>w&ges;1281,1505</f>, and for <f>w≡9,17 (mod 20)</f>, we show that <f>w&ges;2029,2477</f> is sufficient with one possible exceptional series, namely <f>v=4w+9</f> when <f>w≡17 (mod 20)</f>. We know of no example where <f>v=4w+9</f>. 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 <f>w=33</f> and 97, we establish the smallest number of blocks in a pair covering design with <f>k=5</f> when <f>v≡1 (mod 4)</f> with 37 open cases, the largest being for <f>v=489</f>; hitherto, there were 104 open cases, the largest being <f>v=2249</f>. [Copyright &y& Elsevier]