2 research outputs found
On colorings of variable words
In this note, we prove that the base case of the Graham--Rothschild Theorem,
i.e., the one that considers colorings of the (-dimensional) variable words,
admits bounds in the class of Grzegorczyk's hierarchy.Comment: 6 page
A structure theorem for stochastic processes indexed by the discrete hypercube
Let be a finite set with , let be a positive integer,
and let denote the discrete -dimensional hypercube (that is, is
the Cartesian product of many copies of ). Given a family of measurable events in a probability space (a stochastic
process), what structural information can be obtained assuming that the events
are not behaving as if they were independent? We
obtain an answer to this problem (in a strong quantitative sense) subject to a
mild "stationarity" condition. Our result has a number of combinatorial
consequences, including a new (and the most informative so far) proof of the
density Hales--Jewett theorem