A density version of the Carlson-Simpson theorem (Q479520)

The Carlson-Simpson theorem states that, for every integer \(k\geq2\) and every coloring of the set of all words over \(k\), there exist a word \(c\) over \(k\) and a sequence \((w_n)\) of left variable words over \(k\) such that the set \[ \{c\}\cup\{c^\frown w_0(a_0)^\frown\dots^\frown w_n(a_n): n\in\mathbb{N},\;a_0,\dots,a_n\in[k]\} \] is monochromatic. After a short prehistory of the theorem, the authors formulate three density versions and prove them on 60 pages. Among various tools, the Szemeredy technique plays the key part.