Home

# Borel sets and Ramsey's theorem1

## Extract

Definition 1. For a set S and a cardinal κ,

In particular, 2ω denotes the power set of the natural numbers and not the cardinal 2ℵ0. We regard 2ω as a topological space with the usual product topology.

Definition 2. A set S ⊆ 2ω is Ramsey if there is an M ∈ [ω]ω such that either [M]ωS or else [M]ω ⊆ 2ωS.

Erdös and Rado [3, Example 1, p. 434] showed that not every S ⊆ 2ω is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.

The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S1 and S2 are disjoint open subsets of 2ω, there is an M ∈ [ω]ω such that either [M]ωS1 = ∅ or [M]ωS2 = ⊆. (This is halfway between “clopen sets are Ramsey” and “open sets are Ramsey.”) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.

## Footnotes

Hide All
1

The research for this paper was supported by NSF grant GP-27964 and 144-A966.

## References

Hide All
[1]Chang, C. C. and Galvin, Fred, A combinatorial theorem with applications to polarized partition relations (to appear).
[2]Erdös, P. and Hajnal, A., Unsolved problems in set theory, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part 1, American Mathematical Society, Providence, R.I., 1971, pp. 1748.
[3]Erdös, P. and Rado, R., Combinatorial theorems on classifications of subsets of a given set, Proceedings of the London Mathematical Society (3), vol. 2 (1952), pp. 417439.
[4]Galvin, Fred, A generalization of Ramsey's theorem, Notices of the American Mathematical Society, vol. 15 (1968), p. 548. Abstract 68T–368.
[5]Martin, D. A. and Solovay, R. M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143178.
[6]Mathias, A. R. D., On a generalization of Ramsey's theorem, Notices of the American Mathematical Society, vol. 15 (1968), p. 931. Abstract 68T–E19.
[7]Nash-Williams, C. St. J. A.. On well-quasi-ordering transfinite sequences, Proceedings of the Cambridge Philosophical Society, vol. 61 (1965), pp. 3339.
[8]Ramsey, F. P., On a problem of formal logic, Proceedings of the London Mathematical Society (2), vol. 30 (1930), pp. 264286.
[9]Silver, Jack, Every analytic set is Ramsey, this Journal, vol. 35 (1970), pp. 6064.
[10]Soare, Robert I., Sets with no subset of higher degree, this Journal, vol. 34 (1969), pp. 5356.
[11]Solovay, Robert M., A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics (2), vol. 92 (1970), pp. 156.

# Borel sets and Ramsey's theorem1

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *