A Partition Theorem

Abstract

We prove the following: there is a primitive recursive function f_-^*(-,-), in the three variables, such that: for every natural numbers t,n>0, and c, for any natural number k>=f^*_t(n,c) the following holds. Assume L is an alphabet with n>0 letters, M is the family of non empty subsets of {1,...,k} with =<t members and V is the set of functions from M to L, and lastly d is a c-colouring of V (i.e. a function with domain V and range with at most c members). Then there is a d-monochromatic V-line, which means that there are w included in {1,...,k}, with at least t members and a function r from {u in M: u not a subset of w} to L such that letting Y={eta in V: eta extends r and for each s=1,...,t it is constant on {u in M: u is an s-element subset of w}}, we have: the restriction of d to Y is constant (for t=1 those are the Hales Jewett numbers).