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TONY PADILLA: A very very big number, a super super big number.
In fact, it's just an off-the-scale big number, and that's TREE(3).
It absolutely puts Graham's Number to shame.
I mean, really Graham's Number is effectively zero compared to TREE(3).
Let's explain where TREE(3) comes from.
Well, It comes from a game of trees.
There are three different types of seeds.
We're gonna have a green seed...
Mathematicians wouldn't call these seeds,
they'd call them nodes, but we'll call them seeds.
Okay, and a black seed, and a red seed.
And what we're gonna try and do is we're gonna try to build a forest.
Okay, one tree at a time.
The first tree can't have more than one seed.
The second tree can't have more than two seeds.
The third tree can't have more than three seeds and so on, okay?
And that's rule number one.
The other rule is that if you build a tree,
if you find that an earlier tree could've been contained within that tree,
the whole forest dies.
Okay? So let's just sort of illustrate what we mean,
so let's try and build a tree for example, so we might start off with,
you know, a green seed...
And then we branch up and we get a black seed...
And then maybe we have two branches...
And there, you know we can have another green seed maybe...
And we can sort of draw these trees, right?
We said that the first go, the first tree, shouldn't have more than one seed,
the second tree shouldn't have more than two seeds, and so on.
We also said if you build a tree,
then an earlier tree could've been contained within it, then the forest dies,
so what do we mean by contained?
Well the mathematical term that we're really talking about here is inf-embeddable.
Don't worry about what inf-embeddable is,
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