Cal Newport

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.

Here's the best.

Here's the most possible point.

So they didn't replace Erdos' conjecture with a better conjecture.

They didn't prove Erdos' conjecture, but they provided a counterexample construction that showed the thing he thought was the right answer couldn't possibly be right.

Now, how did they do this?

Well, they used a reasoning LLM.

So that's an LLM that has been tuned to essentially talk out loud, to sort of think out loud and wander with its thoughts.

We first saw the first reasoning models back in 2024 with O1 and the O models, a deep sequence reasoning model as well.