The density Hales-Jewett theorem states that for every integer 
and real number 
, every subset 
of 
with 
contains a combinatorial
line once 
is sufficiently large (Furstenberg and Katznelson 1991). A
combinatorial line consists of the 
words obtained by choosing a nonempty set
of coordinate positions, placing the same variable letter in those positions, and
fixing all remaining letters.
Polymath (2012) gave an elementary proof of the theorem.
This is problem 171 in the Erdős problems collection (Bloom 2026).
