Canonical Partition Theorems for Parameter Sets

A canonical (i.e., unrestricted) version of the partition theorem for k-parameter sets of Graham and Rothschild (Trans. Amer. Math. Sot. 159 (1971), 257-291) is proven. Some applications, e.g., canonical versions. of the Rado-Folkman-Sanders theorem and of the partition theorem for finite Boolean algebras are given. Also the ErdBs-Rado canonization theorem (J. London Math. Sot. 25 (1950), 249-255) turns out to be an immediate corollary. A. INTRODUCTION " Classical " partition (Ramsey) theory investigates the behavior of structures with respect to colorings of substructures with only a small number of colors. The main question is whether it is possible to obtain monochromatic (i.e., constantly colored) substructures. For a survey on Ramsey theory see, e.g., [5]. R ecent research considers more general colorings, viz., colorings with an arbitrary number of colors. Of course, generally one cannot expect to find monochromatic substructures. But possibly one always can be restricted to certain types of colorings, for example, structures on which the coloring is either constant or one-to-one. An example of such a theorem is the so-called " canonical version " of van der Waerden's theorem on arithmetic progressions, which is due to Erdos and Graham. THEOREM. Fo.r-ezrery positive integer k there exists a positive integer n such that for every coloring A : {O,..., n-1 ]-+ o of the first n nonnegative integers with arbitrary many colors (i.e., with an infinite number of colors) there extsts an arithmetic progression a, a + d,..., a + (k-1)-d of k terms such that A r {a,..., a + (k-1). d} is either a constant coloring or a one-to-one coloring. However, things do not always behave so nicely; sometimes it is certainly not true that one can always be restricted to a constant or a one-to-one 309 310 PRGMELANDVOIGT coloring. A prototypical result in this direction is the so-called " Erdbs-Rado canonization theorem. " This may be viewed as the generalization of Ramsey's theorem to arbitrary colorings. THEOREM [ 31. Let k, m be positive integers. Then there exists a positive integer n such that for every coloring A : [n] "-+ w of the k-element subsets of n = {O,..., n-1 } with infinitely many colors there exists an m-element subset X of n and a-possibly empty-subset X G {O,..., k-1 } of k such that for any two kIn other words, with respect to colorings of k-element subsets there exist 2k different types of canonical …