The quasi order of graphs on an ordinal

For α an ordinal, a graph with vertex set α may be represented by its characteristic function, f : [ α ] 2 → 2 , where f ( { γ , δ } ) = 1 if and only if the pair { γ , δ } is joined in the graph. We call these functions α - colorings. We introduce a quasi order on the α -colorings (graphs) by setting f ≤ g if and only if there is an order-preserving mapping t : α → α such that f ( { γ , δ } ) = g ( { t ( γ ) , t ( δ ) } ) for all { γ , δ } ∈ [ α ] 2 . An α -coloring f is an atom if g ≤ f implies f ≤ g . We show that for α = ω ω below every coloring there is an atom and there are continuum many atoms. For α < ω ω below every coloring there is an atom and there are finitely many atoms.