Further results on (v,{5,w∗},1)(v,{5,w∗},1)-PBDs

In this article we investigate the existence of pairwise balanced designs on vv points having blocks of size five, with a distinguished block of size ww, briefly (v,{5,w∗},1)(v,{5,w∗},1)-PBDs.The necessary conditions for the existence of a (v,{5,w∗},1)(v,{5,w∗},1)-PBD with a distinguished block of size ww with v>wv>w are that v≥4w+1v≥4w+1, v≡w≡1(mod4) and either v≡w(mod20) or v+w≡6(mod20). Previously, Bennett et al. had shown that these conditions are sufficient for w>2457w>2457 with the possible exception of v=4w+9v=4w+9 when w≡17(mod20), and had studied w≤97w≤97 in detail, showing there that the necessary conditions are sufficient with 71 possible exceptions.In this article, we show sufficiency for w≡1,5,13(mod20) and give a small list of possible exceptions containing 26 and 104 values for w≡9,17(mod20). For w≡9(mod20), all possible exceptions satisfy either v=4w+13v=4w+13 with w≤489w≤489 or v≢w(mod20) with v<5wv<5w and w≤129w≤129; for w≡17(mod20), all possible exceptions except (v,w)=(197,37),(529,37)(v,w)=(197,37),(529,37) satisfy either v=4w+9v=4w+9 with w≤1757w≤1757 or v≢w(mod20) with v<5wv<5w and w≤257w≤257.As an application of our results for w=97w=97, we establish that, if v≡9,17(mod20), v≥389v≥389 and v≠429v≠429, then the smallest number of blocks in a pair covering design with k=5k=5 is ⌈v(v−1)/20⌉⌈v(v−1)/20⌉, i.e., the Schönheim bound.