Partitions of Products of Finite Sets
by
Jimena Llopis
Universidad Simón Bolívar
Coauthors: Stevo Todorcevic (University of Toronto)

In this paper we study partitions of infinite products of discrete spaces. We are interested in finding for a given sequence of natural numbers, a product of finite sets and a large class of partitions such that every partition from the class is constant in some subproduct of the prescribed size (given by the initial sequence of natural numbers). We prove results of the following sort:

For every infinite sequence { m_i }_{i in N} of natural numbers, there is an increasing sequence { n_i }_{i in N} such that for every partition f : (\prod_{i in N} n_i) --> 2 (f in a certain class of functions), there are sets { H_i }_{i in N} with |H_i| = m_i such that f is constant on (\prod_{i in N} H_i).

We first deal with the class of partitions f : (\prod_{i in N} n_i) --> 2 such that f^-1({0}) is a closed subspace of the product \prod_{i in N} n_i. Then by induction we extend the result to a larger class of functions, namely those which divide \prod_{i in N} n_i into two Borel subspaces.

We also relate this type of partitions to partitions with real parameters, meaning partitions whose domains are products of the form (\prod_{i in N} n_i) ×R .

It is interesting that the corresponding parametrized partition theorems, even when replacing the product \prod_{i in N} n_i of finite sets by N^N, are generaly stronger than the partition theorems without parameters for products of finite sets.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caai-65.