The smallest dimension of a hypercube such that if the lines joining all pairs of corners are two-colored, a planar complete graph K_4 of one color will be forced. The actual number is believed to be 6, but the best bound proved is so large that if all the material in the universe was turned into pen and ink it would not be enough to write the number down.
Now, if we build an Internet which has Graham's Number possible addresses, we had better hope that Graham's number isn't actually six, or there's going to be some problems...
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The smallest dimension of a hypercube such that if the lines joining all pairs of corners are two-colored, a planar complete graph K_4 of one color will be forced. The actual number is believed to be 6, but the best bound proved is so large that if all the material in the universe was turned into pen and ink it would not be enough to write the number down.
Now, if we build an Internet which has Graham's Number possible addresses, we had better hope that Graham's number isn't actually six, or there's going to be some problems...