IP* Sets in Product Spaces
by
Neil Hindman
Howard University
Coauthors: Vitaly Bergelson

An IP* set in a semigroup (S, ·) is a set which meets every set of the form

FP( <xn : n in N >) = { \Pin in F xn : F is a finite nonempty subset of N },

where the products are taken in increasing order of indices. We show, using the Stone-Cech compactification of the product space S ×S, that if S is commutative, then whenever C is an IP* set in S ×S, C contains cartesian products of arbitrarily large finite substructures of any given

FP( <xn : n in N >) ×FP( <yn : n in N >)

We show further that C need not contain any infinite substructures and that the commutativity hypothesis may not be omitted. Finally we show, in case S is a (not necessarily commutative) group, that a class of richer sets, known as the Po* sets, do have to contain cartesian products of infinite subsystems-even of subsystems drawn in a uniform fashion from FP( <x_n : n in N >) and FP( <y_n : n in N >)

Date received: April 12, 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 Conferences Inc. Document # caae-37.