A note on a problem of Erdös and Hajnal

AbstractIn [5], Erdös and Hajnal formulate the following proposition, which we shall refer to as Φ: If ϕ is an order-type such that |ϕ| = ω2 but ω2, ω2∗ ≰ ϕ, there is ψ ⩽ ϕ, |ψ| = ω1, such that ω1, ω1∗ ≰ ψ. In [2], we showed that if V = L, then ˥Φ. We do not know if the assumption V = L can be weakened to CH, or if, in fact, Φ is consistent with CH. However, in this note we show that, relative to a certain large cardinal assumption, Φ is consistent with 2ω = ω2, so that ˥Φ is not provable in ZFC alone. Our proof has an interesting model-theoretic consequence, which we mention at the end