Uniformly completely Ramsey sets
Galvin and Prikry defined completely Ramsey sets and showed that the class of completely Ramsey sets forms a σ-algebra containing open sets. However, they used two definitions of completely Ramsey. We show that they are not equivalent as they remarked. One of these definitions is a more uniform property than the other. We call it the uniformly completely Ramsey property. We show that some of the results of Ellentuck, Silver, Brown and Aniszczyk concerning completely Ramsey sets also hold for uniformly completely Ramsey sets. We also investigate the relationships between uniformly completely Ramsey sets, universally measurable sets, sets with the Baire property in the restricted sense and Marczewski sets.
- [AFP] B. Aniszczyk, R. Frankiewicz and S. Plewik, Remarks on (s) and Ramsey-measurable functions, Bull. Polish Acad. Sci. Math. 35 (1987), 479-485.
- [B] J. B. Brown, The Ramsey sets and related sigma algebras and ideals, Fund. Math. 136 (1990), 179-185.
- [E] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163-165.
- [GP] F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, ibid. 38 (1973), 193-198.
- [L] A. Louveau, Une démonstration topologique de théorèmes de Silver et Mathias, Bull. Sci. Math. (2) 98 (1974), 97-102.
- [M] A. W. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, 1984, 201-233.
- [S] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60-64.