Zero-sum flows for Steiner triple systems

Given a $2$-$(v,k,\lambda )$ design, $\mathcal{S}=(\mathcal{X},\mathcal{B})$, a {\it zero-sum $n$-flow} of $\mathcal{S}$ is a map $f: \cal{B} \longrightarrow {\pm 1, \ldots ,\pm (n-1)}$suchthatforanypoint$x\in X$,thesumof$f$ around 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 $(v,\lambda )$, for which a triple system, TS$(v,\lambda )$ exists, there exists one which has a zero-sum $3$-flow, except when $(v,\lambda )\in \{(3,1),(4,2),(6,2),(7,1)\}$ and except possibly when $v \equiv 10\pmod{12}$and$\lambda = 2$\mathrm{.}Wealsogivea$O(\lambda^2v^2)$ 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 $v\equiv 3 \pmod 4$,azero-sum$4$-flowif$v\equiv 3 \pmod 6$andwithazero-sum$5$-flowif$v\equiv 1 \pmod 4$.