### Definable Ramsey and definable Erdös ordinals.

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Given an uncountable cardinal κ with $\kappa ={\kappa}^{<\kappa}$ and ${2}^{\kappa}$ regular, we show that there is a forcing that preserves cofinalities less than or equal to ${2}^{\kappa}$ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein subset...

Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $\u27e8{S}_{\alpha}:\alpha <\omega \u2081\u27e9$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S\in L[r,\u27e8S{*}_{\alpha}:\alpha <\omega \u2081\u27e9]$ whenever each $S{*}_{\alpha}$ is equal to ${S}_{\alpha}$ modulo nonstationary changes, or may have the weaker meaning that $S\in L[r,\u27e8{S}_{\alpha}\cap C:\alpha <\omega \u2081\u27e9]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that S gets exactly...