In 1967, F. Arthur Sherk gave a simple proof that the finite metric planes (of Bachmann and Schmidt) are precisely the affine planes of odd order. Moreover, Sherk’s proof holds for a more general class of incidence structures that do not involve the ‘three-reflection theorem’ whatsoever, and thus yields a beautiful characterisation of the finite affine planes of odd order. By relaxing the first of Sherk’s axioms to ‘every pair of points lies on at most one line’, we can study what we call partial Sherk planes. In this talk, we outline our characterisation of these incidence structures as Bruck nets, in the same vein as Sherk’s result, and what it means for connected combinatorial objects such as mutually orthogonal latin squares.

(Joint work with Joanna Fawcett and Jesse Lansdown)