Shift graphs on precompact families of finite sets of natural numbers

We study graphs defined on families of finite sets of natural numbers and their chromatic properties. Of particular interest are graphs for which the edge relation is given by the shift. We show that when considering shift graphs with infinite chromatic number, one can center attention on graphs defined on precompact thin families. We define a quasi-order relation on the collection of uniform families defined in terms of homomorphisms between their corresponding shift graphs, and show that there are descending ω1ω1-sequences. Specker graphs are also considered and their relation with shift graphs is established. We characterize the family of Specker graphs which contain a homomorphic image of a shift graph.