in keeping with the dual nature of the structure. Projective planes constructed from finite fields (Desarguesian planes) have automorphism groups that act transitively on the points of the plane, so for these planes the point removed to form the dual affine plane is immaterial, the results of choosing different points are isomorphic. However, there do exist non-Desarguesian planes and the choice of point to remove in them may result in non-isomorphic dual affine planes having the same parameters. An affine plane is obtained by removing a line and all the points on that line from a projective plane. Since a projective plane is a self-dual configuration, the dual configuration of an affine plane is obtained from a projective plane by removing a point and all the lines through that point. Hence the name of this configuration. Hypergraph properties It is known that the projective plane of order r-1 exists whenever r-1 is a prime power; hence the same is true for the TPP. The finite projective plane of order r-1 contains r2-r+1 vertices and r2-r+1 edges; hence the TPP of order r-1 contains r2-r vertices and r2-2r+1 edges. The TPP of order r-1 is an r-partite hypergraph: its vertices can be partitioned into r parts such that each hyperedge contains exactly one vertex of each part. For example, in the TPP of order 2, the 3 parts are {1,6}, {2,5} and {3,4}. In general, each of the r parts contains r-1 vertices. Each edge in a TPP intersects every other edge. Therefore, its maximum matching size is 1:.On the other hand, covering all edges of the TPP requires all r-1 vertices of one of the parts. Therefore, its minimum vertex-cover size is r-1:.Therefore, the TPP is an extremal hypergraph for Ryser's conjecture. The minimum fractional vertex-cover size of the TPP is r-1 too: assigning a weight of 1/r to each vertex (which is a vertex-cover since each hyperedge contains r vertices) yields a fractional cover of size (r2-r)/r=r-1. Its maximum fractional matching size of the is r-1 too: assigning a weight of 1/(r-1) to each hyperedge (which is a matching since each vertex is contained in r-1 edges) yields a fractional matching of size (r2-2r+1)/(r-1)=r-1. Therefore:.Note that the above fractional matching is perfect, since its size equals the number of vertices in each part of the r-partite hypergraph. However, there is no perfect matching, and moreover, the maximum matching size