Steiner triple systems with high chromatic index

It is conjectured that every Steiner triple system of order $v\mathrm{\ne }7$ has chromatic index at most $(v+3)\mathrm{/}2$ when $v\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}6)$ and at most $(v+5)\mathrm{/}2$ when $v\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}6)$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)\mathrm{/}2$ for each integer $v \equiv 3 \pmod{6}$suchthat$v \geq 15$, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $v$. Finally, we establish for each order $v\equiv 15(\mathrm{m}\mathrm{o}\mathrm{d}18)$ that there are at least $v^{v^2(1\mathrm{/}6+o(1))}$ non-isomorphic Steiner triple systems with chromatic index at least $(v+3)\mathrm{/}2$ and that some of these systems are cyclic.