Maximal arcs in Steiner systems \(S(2,4,v)\) (Q1394817)

A maximal arc in a Steiner system \(S(2,4,v)\) is a set of \((v+2)/3\) elements, with no three contained in a block. If an \(S(2,4,v)\) contains a maximal arc, then \(v\equiv 4\pmod {12}\). The authors prove that this condition is also sufficient. More precisely, they construct, for any \(v\equiv 4\pmod {12}\), a resolvable \(S(2,4,v)\) containing a triple of maximal arcs, with each pair intersecting in a common point. The methods they use are Denilson's construction of maximal arcs in finite projective planes of even order, the self-orthogonal \(1\)-factorization approach and the difference family approach. An application to the colouring problem is presented.