A partition theorem

by Shelah. [Sh:679]
Scientiae Math Japonicae, 2002
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 Lambda 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 Lambda 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 subseteq {1, ...,k}, with at least t members and function rho from {u in M: u not a subset of w} to Lambda such that letting L= {eta in V: eta extend rho and for each s=1, ...,t it is constant on {u in M:u subseteq w has s members}}, we have d restriction L is constant (for t=1 those are the Hales Jewett numbers).

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