# Steiner triple systems with high chromatic index

## TL;DR

This paper constructs Steiner triple systems with high chromatic index for certain orders, providing new lower bounds and showing the abundance of such systems, including cyclic ones, for specific congruence classes.

## Contribution

It introduces explicit constructions of Steiner triple systems with high chromatic index and demonstrates their abundance and structural properties for orders congruent to 3 mod 6 and 15 mod 18.

## Key findings

- Constructed systems with chromatic index at least (v+3)/2 for v ≡ 3 mod 6, v ≥ 15.
- Maximum number of disjoint parallel classes in these systems is sublinear in v.
- Existence of at least v^{v^2(1/6+o(1))} non-isomorphic systems with high chromatic index for v ≡ 15 mod 18.

## Abstract

It is conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 \pmod{6}$ and at most $(v+5)/2$ when $v \equiv 1 \pmod{6}$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)/2$ for each integer $v \equiv 3 \pmod{6}$ such that $v \geq 15$, 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 $v \equiv 15 \pmod{18}$ that there are at least $v^{v^2(1/6+o(1))}$ non-isomorphic Steiner triple systems with chromatic index at least $(v+3)/2$ and that some of these systems are cyclic.