Independent Sets in Steiner Triple Systems

Abstract

A set of points in a Steiner triple system (STS(\(v\))) is said to be independent if no three of these points occur in the same block. In this paper, we derive for each \(k \leq 8\) a closed formula for the number of independent sets of cardinality \(k\) in an STS(\(v\)). We use the formula to prove that every STS(21) has an independent set of cardinality eight and is, as a consequence, \(4\)-colourable.