Maximum uniformly resolvable designs with block sizes 2 and 4

A central question in design theory dating from Kirkman in 1850 has been the existence of resolvable block designs. In this paper we will concentrate on the case when the block size k = 4 . The necessary condition for a resolvable design to exist when k = 4 is that v ≡ 4 mod 12 ; this was proven sufficient in 1972 by Hanani, Ray-Chaudhuri and Wilson [H. Hanani, D.K. Ray-Chaudhuri, R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972) 343–357]. A resolvable pairwise balanced design with each parallel class consisting of blocks which are all of the same size is called a uniformly resolvable design, a URD. The necessary condition for the existence of a URD with block sizes 2 and 4 is that v ≡ 0 mod 4 . Obviously in a URD with blocks of size 2 and 4 one wishes to have the maximum number of resolution classes of blocks of size 4; these designs are called maximum uniformly resolvable designs or MURDs. So the question of the existence of a MURD on v points has been solved for v ≡ 4 ( mod 12 ) by the result of Hanani, Ray-Chaudhuri and Wilson cited above. In the case v ≡ 8 ( mod 12 ) this problem has essentially been solved with a handful of exceptions (see [G. Ge, A.C.H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5-GDDs, J. Combin. Des. 13 (2005) 222–237]). In this paper we consider the case when v ≡ 0 ( mod 12 ) and prove that a MURD ( 12 u ) exists for all u ≥ 2 with the possible exception of u ∈ { 2 , 7 , 9 , 10 , 11 , 13 , 14 , 17 , 19 , 22 , 31 , 34 , 38 , 43 , 46 , 47 , 82 } .