Extension of partial diagonals of matrices II

1.1 We consider three related problems in this paper**. The first is to find necessary and sufficient conditions under which every partial diagonal of prescribed length of a given matrix can be extended to a diagonal of the matrix (or row diagonal if the matrix is rectangular). Our result may be put in set-theoretic terms and is then seen to be of some interest in transversal theory. The second problem is tha t of determining for which matrices the extension may be performed uniquely. Finally, we consider the more difficult problem of determining those n • n matrices which have the property that, for prescribed t, every partial diagonal of length t can be extended to at most one diagonal. We are able to give a complete answer in the case n ~ 3 t 1. The situation is much more complicated for values of n in the range t + 2 < n ~ 3 t 2, and here we obtain some special results of interest but not a complete classification of the possibilities. 1.2 In this section we lay down our notation and terminology and state such preliminary theorems as we shall subsequently need to use. All matrices considered are finite and, unless otherwise stated, have elements 0, 1. A O-submatrix of a given matrix has all its elements zero. Some economy is achieved in certain proofs in w 4 by the admission of vacuous matrices.