  In this note, we investigate a new model theoretical tree property, called
the antichain tree property (ATP). We develop combinatorial techniques for ATP.
First, we show that ATP is always witnessed by a formula in a single free
variable, and for formulas, not having ATP is closed under disjunction. Second,
we show the equivalence of ATP and $k$-ATP, and provide a criterion for
theories to have not ATP (being NATP).
  Using these combinatorial observations, we find algebraic examples of ATP and
NATP, including pure group, pure fields, and valued fields. More precisely, we
prove Mekler's construction for groups, Chatzidakis' style criterion for PAC
fields, and the AKE-style principle for valued fields preserving NATP. And we
give a construction of an antichain tree in the Skolem arithmetic and atomless
Boolean algebras.
