On the relations of various conjectures on Latin squares and straightening coefficients

Abstract

We discuss the relations among various combinatorial conjectures, to wit: I. A conjecture of J. Dinitz on partial Latin squares, closely related to a conjecture of N. Alon and M. Tarsi on Latin squares. II. A conjecture of the second author on the nonvanishing property of a certain straightening coefficient in the supersymmetric bracket algebra. This conjecture is motivated by the author’s program of extending invariant theory by the use of supersymmetric variables. III. A conjecture of the second author on the exchange property satisfied by sets of bases of a vector space (or more general, for bases of a matroid). IV. A conjecture of J. Kahn which generalizes Dinitz’s conjecture, as well as conjecture III.