The uniformity space of hypergraphs and its applications

Let H = (V,E) be a hypergraph, and let F be a field. A function f : V → F is called stable if for each e € E, the sum of the values of f on the members of e is the same. The linear space consisting of the stable functions, denoted by U(H,F), is called the uniformity space of H over F. The dimension of U(H,F), denoted by udim(H,F), is called the uniformity dimension of H over F. The concept of uniformity space carries over to several (weighted) (hyper)graph-theoretic problems, in which we require that all the sub(hyper)graphs with a specific property have the same weight or size. This is done by defining an appropriate hypergraph whose edges represent all the sub(hyper)graphs having this property. Two such natural problems are: • Let G₁ = (V₁,E₁) and G₂ = (V₂, E₂) be two graphs where G₁ is a subgraph of G₂. A function f : E₂ → F is called stable if all the copies of G₁ in G₂ have the same weight. • Let G = (V, E) be a graph. A function f : V → F is called stable if all the maximal (w.r.t. containment) independent sets of G have the same weight. Clearly, many other problems can be formulated, and their resulting uniformity space can be defined. The purpose of this paper is twofold. The first is to determine (or, alternatively, compute efficiently) the uniformity dimension, and a corresponding basis, of several problems. The other purpose is to show applications of the uniformity space concept to other graph-theoretic problems, such as the determination of the zero-sum mod 2 Ramsey numbers.