$bb(7)$ is already extremely large , but I found a discrepancy in the lower bound:
In this survey the lower bound for $bb(7)$ is given as $$ BB(7) > 10^{10^{10^{10^7}}}, $$ hence four tens in the power tower followed by a 7.
But here there are five tens in the power tower followed by a $7$. Which one is true ?With $6$ states and $3$ symbols, the lower bound is HUGE, but I wonder whether I interprete the notation correctly :
Is the bound here i.e. $$2\uparrow^6 8$$ in Knuth Arrow notation ?
A clarification would be appreciated.
The lower bound for $BB(7)$ is the one given in the initial announcement
$$ BB(7) > 10^{10^{10^{10^{18705352}}}} > 10^{10^{10^{10^{10^7}}}}, $$
as confirmed by Cloudy176 and P. Michel in the links posted therein.
The notation used for $BB(6,3)$ is indeed $2 \uparrow^6 8$. The braces indicate Bowers' Exploding Array Notation, which up to three entries correspond with arrow notation as $\{a, b, c\} = a \uparrow^c b$. Also see the explanation posted below in the link, there some explanation is given using Knuth's arrow notation.