It is known that any non-trivial (r,1)-design on υ varieties (υ ⩾ (r− 1)2 − 1) is extendible; this fact implies the existence of a projective plane of order r − 1. In this paper it is shown that any non-trivial (r, 1)-design on (r − 1)2 − α varieties, where r and α are appropriately bounded, is extendible; hence this fact implies the existence of a projective plane of order r − 1. We also show that, for υ ⩾ (r − 1)2 − 2, any non-trivial (r, 1)-design on υ varieties is extendible.
