Prime number arithmetic progression, such as 3, 5, 7. Longer ones, such as 7,37,67,97, 127, 157. If I remember correctly, there are only 26 arithmetic series, the longest known prime number, so the G-T theorem is still an extraordinary result.
This proof is basically done by graph theory and combinatorial mathematics, based on Szemeredi regular lemma and Szemeredi theorem. Szemeredi himself won the Abel Prize for this theorem, Gowers won the Fields Prize for constructing the bounds of constants in the canonical lemma through harmonic analysis, and Tao also won the Fields Prize for popularizing this theorem. No wonder my boss said that within 10 years, regular referral will be a tool that everyone will use. (In particular, Frieze and Kannan gave a weakened version of the regularity lemma, and graph limit theory has been widely used in theoretical computers. )
The most basic theory about graph limit can be referenced. What indicators can describe the similarity between two graphs? -Zhihu
We are not interested in the specific proof process of G-T theorem. Let me give you a general idea. There is a famous conjecture in Erdos. I have a subset A of natural numbers, the element of which is, if, then there is an arbitrarily long arithmetic progression in A. This conjecture is still inconclusive, and people can't even prove that there is a arithmetic progression of length 3 in A. G-T theorem is a special case of this conjecture, because the sum of reciprocal of all prime numbers is infinite. As for the general situation, who can prove this? I don't think the field has run away.
Szemeredi proved the weakened version. The so-called Szemeredi theorem means that if A is a dense subset of natural numbers, there is an arbitrarily long arithmetic progression in A. Dense means that [N] refers to the set of the first n natural numbers. Note that prime numbers do not satisfy this property (from the prime number theorem), so this theorem is weaker than G-T theorem.
The tool he used was the regular lemma.
Szemeredi's regularity lemma describes that complete chaos is impossible. Give me a big picture, and I can divide it into m parts, which makes it very regular between every two parts. Gals proved that m is at least a tower function.
The graph counting lemma can be obtained by regular lemma. Let me take a triangle as an example. Triangle removal lemma says that if there are not many triangles () in a graph, I can remove quite a few edges () to make the graph without triangles. The triangle in the trilogy can correspond to arithmetic progression of length 3. Basically, by counting lemma and some careful analysis, we can get the Szemeredi theorem.
The core of G-T theorem is to prove the relative Szemeredi theorem. Because the prime set P is sparse in N, the content of this theorem is that if the subset S of N satisfies certain properties (the initial paper must satisfy two conditions, and the latest result is simplified to one) and A is dense in S, then there is a arithmetic progression of arbitrary length in A, and we can regard S as a set of numbers with few prime factors, so that the prime numbers are dense in it. The concrete idea is to prove the counting lemma in the graph under sparse conditions.
However, we can see that this latest result is still far from Ordos conjecture. . Looking forward to the breakthrough of the great god.
I eat out, so I won't use reference paper. In addition, does anyone know why Tao used the theory of "green pottery" to occupy the field, but green didn't? He should be 18' s 4 1, there is no hope at all.