I talked to Ronald Graham last night at the Joint Mathematics Meeting in San Diego and asked him the question here. He said he'd made up Graham's number when talking to Martin Gardner because 1) it was simpler to explain than his actual upper bound, the one that appears in his paper with Rothschild, and 2) it's bigger, so it's still an upper bound!

So, apparently the comment on Wikipedia:

This weaker upper bound for the problem, attributed to an unpublished work of Graham [....]

is a bit misleading, though still technically true. I'll try to fix it in a while.

Nice question!

I talked to Ronald Graham last night at the Joint Mathematics Meeting in San Diego and asked him the question here. He said he'd made up Graham's number when talking to Martin Gardner because 1) it was simpler to explain than his actual upper bound, the one that appears in his paper with Rothschild, and 2) it's bigger, so it's still an upper bound!

So, apparently the comment on Wikipedia:

This weaker upper bound for the problem, attributed to an unpublished work of Graham [....]

is a bit misleading, though still technically true. I'll try to fix it in a while.

Nice question!