Little graham's number, Graham's number and the Graham-conway-number

Question

Sbiis Saibian desbribes on his site in section $3.2.9$ the "little-graham-number" He claims that Graham used this number (much smaller than "Graham's number") in his proof, and Gardner published "Graham's number" and finally he mentions the version with fours instead of threes, called the "Graham-conway-number".

• What is known about the history of these $3$ numbers ?

Answer 1

• Little Graham, or the Graham-Rothschild number, was the original upper bound for the problem in the Ramsey Theory.
• Graham's number, of the Graham-Gardner number, is the upper bound used by Gardner is his article, because the proof was easier to explain using this larger number.
• The Graham-Conway number is just a joke form Conway.