Terence Tao

A good example of this is that there's this old theorem in mathematics called Szemeredi's theorem, proven in the 1970s.

It concerns trying to find a certain type of pattern in a set of numbers that the patterns have made progression.

Things like 3, 5, and 7, or 10, 15, and 20.

And Andrea Zamorelli proved that any set of numbers that are sufficiently big, what's called positive density, has arithmetic progressions in it of any length you wish.

So, for example, the odd numbers have a density of one-half,

and they contain rhythmic progressions of any length.

So in that case, it's obvious because the odd numbers are really, really structured.

I can just take 11, 13, 15, 17.

I can easily find rhythmic progressions in that set.

But Zermattislim also applies to random sets.

If I take the set of odd numbers and I flip a coin for each number, and I only keep

the numbers which for which i got a heads okay so i just flip coins i just randomly take out half the numbers i keep one half so that's a set that has no no patterns at all but just from random fluctuations you will still get a lot of um of ethnic progressions in that set can you prove that there's arithmetic progressions of arbitrary length within a random yes um have you heard of the infinite monkey theorem

Usually, mathematicians give boring names to theorems, but occasionally they give colorful names.

The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each typewriter, they type out text randomly, almost surely one of them is going to generate the entire square root of Hamlet, or any other finite string of text.

It will just take some time, quite a lot of time, actually.

But if you have an infinite number, then it happens.

So basically the thing is that if you take an infinite string of digits or whatever, eventually any finite pattern you wish will emerge.

It may take a long time, but it will eventually happen.

In particular, ethnic progressions of any length will eventually happen

Okay, but you need an extremely long random sequence for this to happen.