The quasi order of graphs on an ordinal

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