I Why is Graham's number a factor of 3?
Connect each pair of geometric vertices of an n-dimensional hypercube to obtain a complete graph on 2n vertices. Colour each of the edges of this graph either red or blue. What is the smallest value of n for which every such colouring contains at least one single-coloured complete subgraph on four coplanar vertices?
Looking at the problem and it's 4 coplanar vertices, it seems it's most likely related to the 3 ways these four vertices can be connected. As an example, given vertices A,B,C,D then the connections would be AB,AC,AD but this is only a guess. It could also be related to the spaces divided up by these vertices like a tetrahedron projected onto a plane with one vertex in the center then connecting lines divide up the plane into 3 regions.
That is a great link - its where I learned what little I know about it.
I made the mistake of catching numberphile on this googling it then watching More you tube posts then finally physics forums.
Grahams number is insane, just picturing the structure is insane, 3|||3 makes a googolplex look small. I would like to know how they know now Tree(3) is so much bigger.
There's a set of videos referenced in the post but like the number itself may take a long time to view and reflect upon before you feel you've found an answer to your question.
Ok so what about G(Tree(3)) vs Tree(G64)?
Perhaps @Mark44 or @fresh_42 can comment here.
My only comment is that the thread title is misleading: "Why is Graham's number a factor of 3?"
Since 3 is a prime, its only integer factors are itself and 1. A better thread title would be "Why is Graham's number a power of 3?
That's cheating ha ha! I think the fact these enormous numbers have been used in proofs and the fact they are finite makes them interesting.
Also the way the arrow notation escalates so wildly after only a couple of terms.
The comparisons of what some the numbers could represent and the fact Grahams leaves physical reality pretty quickly is mind boggling.
As an aside, I guess if we consider TREE(n) and Grahams number G(64) mind-boggling what would we consider Graham's cracker more so?
Understanding is not the right word but at least this has given me a glimpse of how fast these beasts can grow.
Why is it 3? 4 seems to make more sense, sides and vertices.
Most likely 1 and 2 were trivial but 3 produced interesting math results.
Also 4 is a multiple of 2 and so may not have been interesting enough.
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