Signed Hypergraph Designs and Diagonal Forms for Some Incidence Matrices

Abstract

Given a t-uniform hypergraph H on k vertices and an assignment of integers f(T) to the t -subsets T of a v-set X, v ≥ k + t, we give necessary and sufficient conditions for the existence of an assignment of integer multiplicities h(G) to those subhypergraphs G of the complete t-uniform hypergraph on v vertices that are isomorphic to H so that the sum of the integers h(G) over those G that contain T is f(T). Our main theorem is stated in terms of integral matrices. As a consequence of our main theorem, e also determine a diagonal form, and hence the p-rank for all primes p, for the incidence matrix of t-subsets versus subhypergraphs isomorphic to H.