Partitioning pairs of countable sets

摘要: We make the translations of our partitions from [3] in context all countable subsets a fixed uncountable set. A different translation was obtained recently by Velleman [4]. The purpose this paper is to define two-cardinal version one [3]. Theorem. For every set there c :[[A]f0]]2 -* such that, for cofinal U C [AfQ' and A, exist x y that c(x, y) = a. proof will use straightforward generalization shall assume equal some initial ordinal 0, we fix an r: [0]8--+ {O, 1}w rx $ ry y. [Identifying w1 with subset 1}t, let (including finite x) be standard code (tp x, qx), where qx defined recursively on sup as follows assuming each cofinality co, have increasing sequence {ai} converging a: If has maximal element 4 qx(0) 1 qx(i + 1) ry(i), nf . limit ordinal, qx(O) 0 qx(2'(2j 1)) (j), xi =x n ai.] Moreover, one-to-one ex : -c o [0*"0 integer [0]Ro, x(n) 4E x: ex(4) < n}. [Qfo0, A(X , Y) /\(rx, ry), i.e., minimal place reals disagree. Finally, [6]fO cA(x, min(y(A(x, y)) \ sup(x A)), Received editors October 10, 1989 and, revised form, February 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05, 04A20. Research at MSRI supported part NSF Grant DMS-8505550. i) 1991 American Mathematical Society 0002-9939/91 $1.00 $.25 per page