Maximal arcs in Steiner systems S(2,4, v)

Abstract

A maximal arc in a Steiner system S(2,4, v) is a set of elements which intersects every block in either two or zero elements. It is well known that v≡4 ( mod 12) is a necessary condition for an S(2,4, v) to possess a maximal arc. We describe methods of constructing an S(2,4, v) with a maximal arc, and settle the longstanding sufficiency question in a strong way. We show that for any v≡4 ( mod 12) , we can construct a resolvable S(2,4, v) containing a triple of maximal arcs, all mutually intersecting in a common point. An application to the motivating colouring problem is presented.