Diagonal forms for incidence matrices and zero-sum Ramsey theory

We consider integer matrices Nt whose rows are indexed by the t-subsets of an n-set and whose columns are all distinct images of a particular column under the symmetric group Sn. Examples include matrices in the association algebras of the Johnson schemes. Three related problems are addressed. What is the Smith normal form (or a diagonal form) for Nt and the rank of Nt over a field of characteristic p? When does the equation N t x = b have a solution x in integers? When is the vector of all ones in the row space of Nt over the field of characteristic p? Previous work provides answers to these questions when the columns of Nt have at least t “isolated vertices”, but interesting problems arise when this is not the case.