Indecomposable triple systems exist for all lambda

AbstractA triple system (v, 3, λ) is indecomposable if it is not the union of two triple systems (v, 3,λ1), (v, 3, λ2.) with λ = λ1 + λ2. A triple system is simple if it has no repeated blocks. In this paper we show the existence of simple indecomposable triple systems for all v and λ satisfying the necessary conditions with v large. Specifically, for each λ we show that there is a v0(λ) (where v0(λ) = O(λ4)) such that there exists a simple indecomposable triple system (v, 3, λ) for eachv ⩾ v0(λ) with λv(v - 1) ≡ 0 (mod 6) and λ(v - 1) ≡ 0 (mod 2). We then concentrate on the case of λ = 5 and show that v0(5) ⩽ 25