Block Transitive Resolutions of t - designs andRoom

By a resolution of t-designs we mean a partition of the trivial design ? X k of all k-subsets of a v-set X into t ? (v 0 ; k;) designs, where v 0 v. A resolution of t-designs with v = v 0 is also called a large set of t-designs. A Room rectangle R, based on ? X k , is a rectangular array whose non-empty entries are k-sets. This array has the further property that taken together the rows form a resolution of t 1 ?designs, and the columns form a resolution of t 2 ?designs. A resolution of t?designs for ? X k is said to admit G as a block transitive automorphism group if G is k?homogeneous on X, and permutes the t?designs of the resolution among themselves. Some examples of block transitive resolutions of nontrivial t-designs, for t 2, are: 1) an M 11-invariant set of 3-(10, 4, 1) designs, 2) an M 12-invariant set of 4-(11, 5, 1) designs, 3) an M 24-invariant set of 2-(21, 5, 1) designs, 4) a P ?L 2 (2 s)-invariant set of 3-(2 s , 4, 1) designs (s = 3 or 5), 5) a P ?L 2 (32)-invariant set of 2?(16; 4; 1) designs, and 6) a variety of P SL 2 (q)-invariant sets of 2-designs with k = 3. We show that this is a complete list. In particular there are no block transitive large sets of t-designs. Moreover, if 1 6 = a < b < c are odd integers such that gcd (a; b) = 1 and ab divides c, then we construct a block transitive Room rectangle based on the 3-subsets of a (7 c + 1)-set whose rows are Steiner triple systems on 7 a points, and whose columns are Steiner triple systems on 7 b points.