Partial Steiner Triple Systems with Equal-Sized Holes

Let 5f be a finite set of x elements, and let G = { G1, G2, ..., Gs} be a par t i t ion of ~r into subsets cal led groups. Let N be a col lect ion of subsets of X cal led blocks, and let set i f = {[B] : B s N ' } , be the set of block sizes. If (~r, N ) has the p r o p e r t y tha t every pa i r of e lements ei ther appears in exact ly one b lock or in exact ly one group , it is a group divisible design and is deno ted by X G D D . The type of the G D D is deno ted by g'llg~g'33.,, g~ when the n u m b e r of g roups of size gi is ti. In this p a p e r we es tabl ish necessary and sufficient condi t ions for the existence of a { 3 } G D D of type url t. W h e n r = 0 , such a { 3 } G D D is jus t a Steiner t r iple system of o rde r t, and when t = 0, such a { 3 } G D D has type u r. In bo th cases, existence has been set t led (see [ 3 ] ) , so we assume tha t r and t are posi t ive th roughout . In par t icu la r , we prove the fol lowing result.