The Spectrum of Tetrahedral Quadruple Systems

An ordered analogue of quadruple systems is tetrahedral quadruple systems. A tetrahedral quadruple system of order v and index λ, TQS(v, λ), is a pair $${(S, \mathcal{T})}$$ where S is a finite set of v elements and $${\mathcal{T}}$$ is a family of oriented tetrahedrons of elements of S called blocks, such that every directed 3-cycle on S is contained in exactly λ blocks of $${\mathcal{T}}$$. When λ = 1, the spectrum problem of TQS(v, 1) has been completely determined. It is proved that a TQS(v, λ) exists if and only if λ(v − 1)(v − 2) ≡ 0 (mod 3), λv(v − 1)(v − 2) ≡ 0 (mod 4) and v ≥ 4.