LIST A: PAPERS ON INFINITARY RAMSEY THEORY, and my dissertation of 1968
[A1] Happy families, Annals of Mathematical Logic 12 (1977) 59--111; MR 58 \# 10462
(Substantially extends the results of my Cambridge dissertation of 1968. §0 introduces the notion of a happy family, (called "selective co-ideal" by later, more prosaic, authors).
Ramsey ultrafilters are those ultrafilters on ω (= {0, 1, 2, …}) which are also happy. In §1 a forcing-free proof is given of the theorem that if F is a Ramsey ultrafilter
and D is an analytic family of infinite subsets of ω, there is an X ∈ F such that for every infinite subset Y of X, Y ∈ D ⇔ X ∈ D. In §2, a notion of forcing associated to a
Ramsey ultrafilter is studied in detail; the notion of a condition capturing a dense set is defined, and used to establish the Mathias and Prikry properties for this forcing; which in turn
lead to proofs using forcing of the above Silver--Mathias theorem. In §3, some general facts about forcing are reviewed, and the Lévy model, obtained by collapsing an inaccessible, and its
celebrated Solovay sub-model are studied. In §4, it is shown that many properties of Ramsey families will hold for all happy families; it follows that no MAD family (i.e. a maximal
infinite family of pairwise almost disjoint infinite subsets of ω)
is analytic. In §5 it is shown that in the Solovay sub-model, all sets of reals are Ramsey, and that had one started from a Mahlo cardinal,
there would be no MAD family there. In §6 a general theorem about Borel functions being smooth is proved; and applied in §7 to prove that under ADR,
there are no MAD families. §8 gives further properties of Mathias reals; in §9, moderately happy families (related to p-points much as happy families
are related to Ramsey ultrafilters) are briefly studied.)
[A2] A remark on rare filters,
Colloquium Mathematicae Societatis Janos Bolyai Volume 10,
(North Holland, 1974), 1095-1097; MR 51 # 10098
(Introduces the idea of a feeble filter, and gives short proofs that if there is a non-feeble filter, then not all sets are Ramsey;
all analytic filters are feeble; and hence no analytic filter is rare, a result first proved using the smooth functions of §6 of [A1];
the present proof incorporates simplifications suggested by Baumgartner.)
[A3]
(with J. M. Henle and H. Woodin) A barren extension,
Proceedings of the VIth Latin American Logic Colloquium, Caracas, 1983,
edited by Carlos Di Prisco, Springer
Lecture Notes in Mathematics, 1130, 195-207; MR 87d:03141
(Proves that if all sets of reals are Ramsey, then a familiar forcing
extension for adding a Ramsey ultrafilter adds no new sets of ordinals; hence, under the further assumption that AD and
V=L(ℝ) hold in the ground model, all strong partition properties of ordinals below Θ remain
true in the extension, but the existence of the ultrafilter
means that AD has become false.)
FROM MY ARCHIVES
On a generalisation of Ramsey's theorem
Introduction
1.
2.
3.
4.
5.
6.
7.
8.
9.
(my Peterhouse Fellowship (1969) and Cambridge Ph.D. (1970) dissertation, written in July 1968 in Bonn-Beuel; the "paragraphs" (really, the chapters) are scanned separately as .pdf files.)
(Introduction: what are now called non-Ramsey sets were then called Scott families. At the time I wrote the thesis, I knew of Prikry's result that all Borel sets are Ramsey, but not that Galvin had also proved it.
¶¶ 1, 2, and 7 are largely devoted to expounding, in the language of Boolean-valued models,
Solovay's construction of his celebrated model, his own account not at the time being available. ¶¶ 3, 4, 5, 6, 8 and 9 are original to me, except where stated. The key concept of a condition capturing a dense set is introduced in ¶ 4, on page 66, and applied in pages 72-77 to prove the ``all subsets generic" property for "plain Mathias" forcing, and later, on page 80, to establish the Prikry property for this forcing. ¶¶ 5 and 6 digress from the main proof; ¶ 5 explores properties of Mathias reals, and has recently borne fruit by showing, answering a question of Zapletal, that if R is an analytic equivalence relation on the power set of omega in which every equivalence class is countable, then,
``almost everywhere", xRy implies that the symmetric difference of x and y is finite.
¶ 6 compares Mathias reals with other kinds of generic real known at the time.
¶ 7 analyses collapsing algebras; in ¶ 8 it is shown that in Lévy's model obtained by collapsing an inaccessible all definable families are Ramsey, and then in ¶ 9 MacAloon's method is followed to obtain "all sets Ramsey" in an appropriate submodel.
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