Claims

../

1970_01_01_erdos_hajnal_milner: Theorem 7 of Erdős, Hajnal and Milner (Balatonfüred colloquium, 1970): a graph with no infinite path on an ordered set of type omega theta below omega_1^(omega+2) has an independent set of the same type.

1986_04_01_larson: Larson (J. London Math. Soc., 1986) proved that if there is no scale of order type omega_1 under eventual domination, then omega_1^(omega+2), the first ordinal not settled in ZFC, has the property.

1987_09_01_larson: Larson (Trans. Amer. Math. Soc., 1987) proved under GCH that for each n at least 2, cofinally many ordinals below omega_n carry a graph with no infinite path and no independent set of their own order type.

1990_04_01_baumgartner_larson: Baumgartner and Larson (Ann. Pure Appl. Logic, 1990) proved that Jensen's diamond gives, on every ordinal below omega_2, a graph with no infinite path and no independent set of type omega_1^(omega+2); every such limit fails.

1990_04_01_larson: Larson (Ann. Pure Appl. Logic, 1990) proved that under Martin's axiom every limit ordinal below the continuum has the property: a graph on it has an infinite path or an independent set of its order type.

---