Partition Theorems for Left and Right Variable Words

In 1984 T. Carlson and S. Simpson established an infinitary extension of the Hales–Jewett Theorem in which the leftmost letters of all but one of the words were required to be variables. (We call such words left variable words.) In this paper we extend the Carlson–Simpson result for left variable words, prove a corresponding result about right variable words, and determine precisely the extent to which left and right variable words can be combined in such extensions. The results mentioned so far all involve a finite alphabet. We show that the results for left variable words do not extend to words over an infinite alphabet, but that the results for right variable words do extend.