# Partial triple systems and edge colourings

## Abstract

A partial triple system of order v, PT(v), is pair (V, B) where V is a v-set, and B is a collection of 3-subsets of V (called triples) such that each 2-subset of V is contained in at most one triple. A maximum partial triple system of order v, MPT(v), is a PT(v), (V, B), such that for any other PT(v), (V, C), we have |C| {slanted equal to or less-than}|B|. Several authors have considered the problem of embedding PT(v) and MPT(v) in systems of higher order. We complete the proof, begun by Mendelsohn and Rosa [6], that an MPT(u) can be embedded in an MPT(v) where v is the smallest value in each congruence class mod 6 with v ≥ 2u. We also consider a general problem concerning transversals of minimum edge-colourings of the complete graph. © 1986.