Cal Newport

All right, so we're looking at this unit distance, planar unit distance conjecture.

Erdos was convinced that he had identified the answer to the question.

I don't want to get too mathy here, but just to say it quickly, Erdos thought that if you were placing endpoints into the plane,

the maximum number of points that you could get to be a unit distance apart would be upper bounded by n raised to the power of 1 plus some constant c divided by the double log of n. Now, as you're probably noticing as you listen to me, that second term in the sum is going to tend towards 0 as n increases, asymptotically speaking.

So this result, the answer...

asymptotically is going to approach plain linear as the point set increases.

That's a really elegant answer.

Erdos was convinced that was right.

A lot of other mathematicians just assumed that that was right because Erdos is usually right.

And so people tried for a long time to prove that was indeed the fundamental limit.

Now, what OpenAI did was they released a paper that said, no, that's wrong.

We actually have a counterexample.

We have a way of placing points that

That is going to have more points.

We're going to feature more points at unit distance than that limit, even as n increases.

I believe the actual bound is something like n plus 1 plus some small fixed constant epsilon that stays fixed as you increase n as opposed to approaching 0.

So they had a counterexample.

They didn't say...

Here's the right answer.

Here is what the limit is.