Zero-sum flows for triple systems

Given a 2-(v,k,λ) design, S=(X,B), a zero-sum n-flow of S is a map f:B⟶{±1,…,±(n−1)} such that for any point x∈X, the sum of f over all the blocks incident with x is zero. It has been conjectured that every Steiner triple system, STS(v), on v points (v>7) admits a zero-sum 3-flow. We show that for every pair (v,λ) for which a triple system, TS(v,λ), exists, there exists one which has a zero-sum 3-flow, except when (v,λ)∈{(3,1),(4,2),(6,2),(7,1)}. We also give a O(λ2v2) bound on n and a recursive result which shows that every STS(v) with a zero-sum 3-flow can be embedded in an STS(2v+1) with a zero-sum 3-flow if v≡3(mod4), a zero-sum 4-flow if v≡3(mod6) and with a zero-sum 5-flow if v≡1(mod4).