Simple signed Steiner triple systems

TL;DR

This paper characterizes the existence of simple signed Steiner triple systems, a combinatorial design where triples are signed to balance pair occurrences, providing necessary and sufficient conditions based on parameters v and s.

Contribution

It establishes the exact conditions under which simple signed Steiner triple systems exist, filling a gap in combinatorial design theory.

Findings

Existence characterized for all relevant parameters v and s.

Provides explicit non-existence for certain small cases like v=7.

Defines the range of s for which ST(v,s) exists based on v.

Abstract

Let $X$ be a $v$-set, $\B$ a set of 3-subsets (triples) of $X$, and $\B^+\cup\B^-$ a partition of $\B$ with $|\B^-|=s$. The pair $(X,\B)$ is called a simple signed Steiner triple system, denoted by ST$(v,s)$, if the number of occurrences of every 2-subset of $X$ in triples $B\in\B^+$ is one more than the number of occurrences in triples $B\in\B^-$. In this paper we prove that $\st(v,s)$ exists if and only if $v\equiv1,3\pmod6$, $v\ne7$, and $s\in\{0,1,...,s_v-6,s_v-4,s_v\}$, where $s_v=v(v-1)(v-3)/12$ and for $v=7$, $s\in\{0,2,3,5,6,8,14\}$.