Greedy clique sequence

Greedy clique sequences are a concept from graph theory that lead to fast-growing functions. Defined by Harvey Friedman in 2010, the resulting functions are some of the strongest Friedman has ever defined.

Definition
Given a positive integer \(k\), let \(G\) be a simple undirected graph (that is, no loops or duplicate edges) with \(\mathbb{Q}^k\) as vertices. We define a sequence \(x\) as a nonempty tuple \((x_1, x_2, \ldots, x_n)\) where \(x_i \in \mathbb{Q}^k\).

\(x\) is an upper shift greedy clique sequence iff:


 * 1) \(x_1\) is only 0's.
 * 2) Let \(m\) be a positive integer such that \(2 \leq 2m \leq n - 1\). Define \(Y\) as the concatenation \(\langle x_1,x_2,\ldots, x_{2m-1}\rangle\) (which is in \(Q^{(2m-1)k}\)) and let \(y\) be the size-\(k\) "substring" of \(Y\) starting at position \(m\). Then \(x_{2m} = y \vee (x_{2m} < y \wedge (y, 2m) \text{ is not an edge of } G)\). If \(x_{2m+1}\) is the upper shift of \(x_{2m}\) [sic].
 * 3) \(\{x_2, x_3, \ldots, x_n\}\) is a  in \(G\). That is, \(G\) contains as an edge every pair of vertices in \(\{x_2, x_3, \ldots, x_n\}\).

\(x\) is an upper shift greedy down clique sequence iff:


 * 1) \(x_1\) is only 0's.
 * 2) Let \(m\) be a positive integer such that \(2 \leq 2m \leq n - 1\). Define \(Y\) as the concatenation \(\langle x_1,x_2,\ldots, x_{2m-1}\rangle\) (which is in \(Q^{(2m-1)k}\)) and let \(y\) be the size-\(k\) "substring" of \(Y\) starting at position \(m\). Then \(x_{2m} = y \vee (x_{2m} < y \wedge (y, 2m) \text{ is not an edge of } G)\). If \(x_{2m+1}\) is the upper shift of \(x_{2m}\) [sic].
 * 3) \(\{x_2, x_3, \ldots, x_n\}\) is a  in \(G\). That is, \(G\) contains as an edge every pair of vertices in \(\{x_2, x_3, \ldots, x_n\}\).