Classification of partitions of all triples on ten points into copies of Fano and affine planes Original Research Article

We continue our study of partitions of the full set of image triples chosen from a image-set into copies of the Fano plane image (Fano partitions) or copies of the affine plane image (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are image where Fano partitions exist, image where affine partitions exist, and image where both affine and mixed partitions exist. The Fano partitions for image and the affine partitions for image and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for image, and (2) the mixed partitions for image into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of image.