A simple proof of the density Hales-Jewett theorem
  We give a purely combinatorial proof of the density Hales--Jewett Theorem that is modeled after Polymath's proof but is significantly simpler. In particular, we avoid the use of the equal-slices measure and work exclusively with the uniform measure. 
Pandelis Dodos, Vassilis Kanellopoulos and Konstantinos Tyros
math.CO