Finite promise games and greedy clique sequences

The finite promise games are a collection of mathematical games developed by American mathematician Harvey Friedman in 2009 which are used to develop a family of fast-growing functions ${\displaystyle FPLCI(k)}$, ${\displaystyle FPCI(k)}$ and ${\displaystyle FLCI(k)}$. The greedy clique sequence is a graph theory concept, also developed by Friedman in 2010, which are used to develop fast-growing functions ${\displaystyle USGCS(k)}$, ${\displaystyle USGDCS(k)}$ and ${\displaystyle USGDCS_{2}(k)}$.

${\displaystyle {\mathsf {SMAH}}}$ represents the theory of ZFC plus, the infinite family of axioms "there exists a strongly ${\displaystyle k}$-Mahlo cardinal for all positive integers ${\displaystyle k}$. and ${\displaystyle {\mathsf {SMAH^{+}}}}$ represents the theory of ZFC plus "for each ${\displaystyle k}$, there is a strongly ${\displaystyle k}$-Mahlo cardinal". ${\displaystyle {\mathsf {SRP}}}$ represents the theory of ZFC plus, for each ${\displaystyle k}$, "there is a ${\displaystyle k}$-stationary Ramsey cardinal", and ${\displaystyle {\mathsf {SRP^{+}}}}$ represents the theory of ZFC plus "for each ${\displaystyle k}$, there is a strongly ${\displaystyle k}$-stationary Ramsey cardinal". ${\displaystyle {\mathsf {HUGE}}}$ represents the theory of ZFC plus, for each ${\displaystyle k}$, "there is a ${\displaystyle k}$-huge cardinal", and ${\displaystyle {\mathsf {HUGE^{+}}}}$ represents the theory of ZFC plus "for each ${\displaystyle k}$, there is a strongly ${\displaystyle k}$-huge cardinal".