Embedding Finite Partial Linear Spaces in Finite Translation Nets

In the 1970’s Paul Erdős and Dominic Welsh independently posed the problem of whether all finite partial linear spaces L are embeddable in finite projective planes. Except for the case when L has a unique embedding in a projective plane with few additional points, very little has been done which is directly applicable to this problem. In this paper it is proved that every finite partial linear space L is embeddable in a finite translation net generated by a partial spread of a vector space of even dimension. The question of whether every finite partial linear space is embedded in a finite André net is also explored. It is shown that for each positive integer n there exist finite partial linear spaces which do not embed in any André net of dimension less than or equal to n over its kernel. Mathematics Subject Classification (2000): 51E15, 51E26