Sets of type-(1,n) in biplanes

A set of type-(m,n) S is a set of points of a design with the property that each block of the design meets either m points or n points of S. If m=1, S gives rise to a subdesign of the design. The parameters of sets of type-(1,n) in finite projective planes were characterised by G. Tallini and M. Tallini Scafati with more generalised order condition. It follows from their result that, a set of type-(1,n) exists only in the planes of square orders and it gives rise to either a Baer subplane or a unital of a finite projective plane of square order. In this paper, we characterise the parameters of sets of type-(1,n) in biplanes of more extended order condition than prime power. It follows from the results that a set of type-(1,n) in a biplane is either a Baer subdesign, a Hermitian subdesign or a subdesign with certain types of parameters. In addition, some examples of sets of type-(1,n) are given in known biplanes.