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-((₂ⁿ), 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-((₂ⁿ), K, λ) design with n 2t + 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-((₂ⁿ), K, λ) designs with n 2t + λ + 4.

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