Bounding graphical t-wise balanced designs

Abstract

Let X be the edges of the complete graph K n on n vertices, provided with the natural action of S n , the automorphism group of K n . A t-wise balanced design ( X, B) with parameters t-(( 2 n ), K, λ) is said to be graphical if B is fixed under the action of S n . We show that for any pair ( t, λ) with t > 1 or λ odd, there cannot exist a non-trivial graphical t-(( 2 n ), K, λ) design with n ⩾ 2 t + λ + 4. Thus, in particular, for each such pair ( t, λ) there are only a finite number of non-trivial graphical t-( v, K, λ) designs. If we further assume no repeated blocks, then for all cases with t > 1 or λ ≠ 2, there do not exist non-trivial graphical t-(( 2 n ), K, λ) designs with n ⩾ 2 t + λ + 4.