The Fine Structure of $(v,3)$ Directed Triple Systems: $v\equiv 0$ or $1(mod3)$

Abstract

The fine structure of a directed triple system of index $\lambda$ is the vector $(c_1,c_2,\dots ,c_{\lambda })$, where $c_i$ is the number of directed triples appearing precisely $i$ times in the system. We determine necessary and sufficient conditions for a vector to be the fine structure of a directed triple system of index $3$ for $v\equiv 0$ or $1(\mathrm{mod}3)$.