Simple Signed Steiner Triple Systems

Let X be a v ‐set, B be a set of 3‐subsets (triples) of X , andB + ∪ B −be 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 ∈ B +is one more than the number of occurrences in triples B ∈ B − . In this paper, we prove that ST ( v , s ) exists if and only if v ≡ 1 , 3( mod 6 ) , v ≠ 7 , and s ∈ { 0 , 1 , . . . , s v − 6 , s v − 4 , s v } , wheres v = v ( v − 1 ) ( v − 3 ) / 12 and for v = 7 , s ∈ { 0 , 2 , 3 , 5 , 6 , 8 , 14 } . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012