[ Q(x) = \sum_i<j (x_i - x_j)^2 ]
Prior proofs gave extremely weak bounds (e.g., Ackermann-type or tower-of-exponentials). Polymath 6.1 sought to reduce the tower height.
For precise algebraic form, consult the (section “Key lemma” or “Key polynomial”) or the final paper: “Density Hales-Jewett and Moser numbers” (2012). polymath 6.1 key
[ \textKey function: f(x) = \text(# of 0's) - \text(# of 1's) \quad \textmod something? ]
Existing approaches involved iterating a “density increment” step, but each step reduced the dimension dramatically. The key polynomial helped track density increments more efficiently. 4. Specifics of the “Key Polynomial” While Polymath 6.1 did not name one single polynomial “the key,” the following polynomial (or its variants) played the central role: [ Q(x) = \sum_i<j (x_i - x_j)^2 ]
or more combinatorially:
[ P(\mathbfx) = \sum_i=1^n \omega^x_i \quad \text(where $\omega$ is a primitive 3rd root of unity) ] [ \textKey function: f(x) = \text(# of 0's)
Let $x_1, x_2, \dots, x_n$ be variables in $0,1,2$ (or $\mathbbF_3$). Consider: