Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on vv points having blocks of size 4, briefly (v,4,1)(v,4,1) RBIBDs. The problem we investigate is the existence of (v,4,1)(v,4,1) RBIBDs containing a (w,4,1)(w,4,1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w)IRB(v,w). The necessary conditions for the existence of an IRB(v,w)IRB(v,w) are that v⩾4wv⩾4w and v≡w≡4(mod12). We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever w≡16(mod60) and whenever w⩾1852w⩾1852.We also give some results on pairwise balanced designs on vv points containing (at least one) block of size ww, i.e., a (v,{K,w*},1)(v,{K,w*},1)-PBD.If the list of permitted block sizes, K5K5, contains all integers of size 5 or more, and v,w∈K5v,w∈K5, then a necessary condition on this PBD is v⩾4w+1v⩾4w+1. We show this condition is not sufficient for any w⩾5w⩾5 and give the complete spectrum (in vv) for 5⩽w⩽85⩽w⩽8, as well as showing the condition v⩾5wv⩾5w is sufficient with some definite exceptions for w=5w=5 and 66, and some possible exceptions when w=15w=15, namely 77⩽v⩽7977⩽v⩽79. The existence of this PBD implies the existence of an IRB(12v+4,12w+4)IRB(12v+4,12w+4).If the list of permitted block sizes, K1(4)K1(4), contains all integers ≡1(mod4), and v,w∈K1(4)v,w∈K1(4), then a necessary condition on this PBD is v⩾4w+1v⩾4w+1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w⩾1037w⩾1037. The existence of this PBD implies the existence of an IRB(3v+1,3w+1)(3v+1,3w+1).