Some Partitions of Three-Dimensional Combinatorial Cubes

When studying certain class of structures (e.g., sets of reals) one frequently enters the following situation. We are given an element A of the class and a mapping f defined on some cube A r o r [ A ] r of A and we need to find a large B _ A where the behaviour of f is as simple as possible. Here is a sample of results of this sort. There is c: [Q]2 ~ {0, 1} such that c"[P] 2 = {0, 1} for every converging sequence (together with its limit) P _c Q, but for every f : [Q]2 ~ {0 , . . . , k} there is a converging sequence P _ Q such that If"[P]L -< 2 (see [0]). There is an uncountable set X and c: X s ~ {0, 1} such that c " A × B × C = {0, 1} for every infinite A, B, C __X, but for every uncountable set X and f : X 3 ~ {0 , . . . , k} there exist infinite A, B, C ___ X such that l f " A × B × C] < 2 (see [14]). There is a Borel map c: [I~] 4 --~ {0 . . . . . 5} such that c"[P] 4 = {0 . . . . . 5} for every perfect set P _ N but for every Borel map f : [N]4 ~ {0 . . . . , k} there is a perfect set P _c R such that ]f,,[p]4] _< 6 (see [1]). There is c: [Q]5 ___, {0, . . . ,7935} such that c " [ X 5] = {0, . . . ,7935} for every X _ Q orderisomorphic to Q, but for every f : [Q]5 ~ {0 . . . . , k} there is X _c ~ orderisomorphic to Q such that If"[X]SI _< 7936 (see [3]). One of the most famous (and potentially useful) problems of this sort asks whether there is a similar behaviour of functions of the form f : [ ~ ] r ~ {0 , . . . , k} on uncoun tab le subcubes [B] r of [N]E To make this problem precise let us introduce the following variation of the usual partition symbol. The symbol a ~ ( b ) ~ / l denotes the s tatement that for every f : [a] r -~ k there exists B __ a of size b such that ]f"[B] ~ < I. Let a --* [b]~+ 1 denote the statement a ~ (b)rk+l/k . Thus the negation a -~ [b]~ is the statement that there is f : [a] r --* k such that f " [ B ] r = k for every B _ a of size b. (We shall be interested mostly in finite k but note that this negation makes sense for any cardinal k.) In this notation the general problem about the