The combinatorics of solving linear equations

2. The combinatorics of solving linear equations Origins: (I. Schur) Does the congruence always have non-trivial solutions? 1875 - 1941

3. The combinatorics of solving linear equations Origins: (I. Schur) Does the congruence always have non-trivial solutions? 1875 - 1941 Yes ! Theorem (Schur – 1916) If the positive integers are partitioned into finitely many classes then at least one of the classes contains solutions to the equation

4. [r] into color classes Some notation: - an r-coloring of induces a partition of Objects in a single color class will be called monochromatic.

5. Schur’s theorem restated: Schur’s theorem restated: In any r-coloring of there is always a monochromatic solution to the equation

6. B. L. van der Waerden (1903 – 1996) Schur’s theorem restated: Schur’s theorem restated: In any r-coloring of there is always a monochromatic solution to the equation Theorem (van der Waerden – 1927) In any r-coloring of there is always a monochromatic arithmetic progression of length k (= k-AP).

7. B. L. van der Waerden (1903 – 1996) Schur’s theorem restated: Schur’s theorem restated: In any r-coloring of there is always a monochromatic solution to the equation Theorem (van der Waerden – 1927) In any r-coloring of there is always a monochromatic arithmetic progression of length k (= k-AP). Special case k = 3. In any r-coloring of there is always a monochromatic solution to the equation

8. What about the equation

9. What about the equation Fact. There is a 4-coloring of with no monochromatic solution to

10. What about the equation Fact. There is a 4-coloring of with no monochromatic solution to [4] by setting Proof. Define Then which is impossible.

11. In general, call an equation r-regular if every r-coloring of contains a monochromatic solution to E. Also, call E regular if it is r-regular for every r.

12. E is regular iff it can be solved with all and not all 0. In general, call an equation r-regular if every r-coloring of contains a monochromatic solution to E. Also, call E regular if it is r-regular for every r. Which equations are regular ? Theorem (R. Rado – 1933)

13. E is regular iff it can be solved with all and not all 0. In general, call an equation r-regular if every r-coloring of contains a monochromatic solution to E. Also, call E regular if it is r-regular for every r. Which equations are regular ? Theorem (R. Rado – 1933)

14. Conjecture (Rado – 1933) For every k, there are equations which are k-regular but are not (k+1)-regular.

15. Conjecture (Rado – 1933) For every k, there are equations which are k-regular but are not (k+1)-regular. Theorem (Alexeev-Tsimerman – 2009) The equation is k-regular but not (k+1)-regular.

16. let denote a system of homogenous linear equations. More generally, for an m x n integer matrix A, For example, for the corresponding system is: which corresponds to an arithmetic progression of length 5 (5-AP).

17. (ii) can be expressed as a rational linear A matrix A is said to satisfy the columns condition (CC) if it possible to partition the columns of A into blocks so that: (i) combination of the columns of

18. (ii) can be expressed as a rational linear A matrix A is said to satisfy the columns condition (CC) if it possible to partition the columns of A into blocks so that: (i) combination of the columns of For example, has CC since

19. The system is regular iff A satifies the columns condition. Theorem (Rado – 1933) Paul Erdös and Richard Rado 1906-1989 1913-1996

20. Call a subset large if for any r, any r-coloring of S always has monochromatic solutions for every regular system. Rado also had many other conjectures concerning systems of regular equations, one of which was this.

21. Call a subset large if for any r, any r-coloring of S always has monochromatic solutions for every regular system. into finitely many parts, at least one of the parts is large. Rado also had many other conjectures concerning systems of regular equations, one of which was this. Conjecture. For any partition of a large set

22. Call a subset large if for any r, any r-coloring of S always has monochromatic solutions for every regular system. into finitely many parts, at least one of the parts is large. Rado also had many other conjectures concerning systems of regular equations, one of which was this. Conjecture. For any partition of a large set This was finally proved in 1973 by Deuber. Walter Deuber – (1943 – 1999)

23. In any r-coloring of there is always a monochromatic arithmetic progression of length k (= k-AP). Even though and are both regular, there is a fundamental difference between them. Recall van der Waerden’s theorem: Erdös and Turán (1936) ask: Which color class has the k-AP’s?

24. In any r-coloring of there is always a monochromatic arithmetic progression of length k (= k-AP). Even though and are both regular, there is a fundamental difference between them. Recall van der Waerden’s theorem: Erdös and Turán (1936) ask: Which color class has the k-AP’s? They conjectured that the “densest” class should have them. More precisely, they conjectured that if satisfies then S should contain k-AP’s for every k.

25. Conjecture ( Erdös -Turán – 1936) For all k, Equivalently, define to be the size of the largest subset of [n] which contains no k-AP.

26. Conjecture ( Erdös -Turán – 1936) For all k, Equivalently, define to be the size of the largest subset of [n] which contains no k-AP. It became clear that even was not going to be so simple to determine because of: Theorem (Behrend – 1946) For a suitable constant c > 0,

27. Conjecture ( Erdös -Turán – 1936) For all k, This shows that for every Equivalently, define to be the size of the largest subset of [n] which contains no k-AP. It became clear that even was not going to be so simple to determine because of: Theorem (Behrend – 1946) For a suitable constant c > 0,

28. The first non-trivial upper bound was given by Roth: Theorem (Roth – 1954)

29. The first non-trivial upper bound was given by Roth: Theorem (Roth – 1954) This was subsequently improved by Szemerédi and Heath-Brown to with the current record being held by Bourgain (2008)

30. The first non-trivial upper bound was given by Roth: Theorem (Roth – 1954) This was subsequently improved by Szemerédi and Heath-Brown to with the current record being held by Bourgain (2008) There has also been a very recent breakthrough improvement of Behrend’s lower bound by Michael Elkin.

31. Behrend (1946) M. Elkin (2009)

32. Behrend (1946) M. Elkin (2009) Is this close to the “truth” ?

33. What about ?

34. What about ? Szemerédi - 1969

35. What about ? Szemerédi - 1969 + $1000 Szemerédi - 1974 Endre Szemerédi – (1940 - )

36. What about ? Szemerédi - 1969 + $1000 Szemerédi - 1974 Endre Szemerédi – (1940 - ) Furstenberg – 1979 (ergodic theory)

37. What about ? Szemerédi - 1969 + $1000 Szemerédi - 1974 Endre Szemerédi – (1940 - ) Furstenberg – 1979 (ergodic theory) Gowers - 1998

38. What about ? Szemerédi - 1969 + $1000 Szemerédi - 1974 Endre Szemerédi – (1940 - ) Furstenberg – 1979 (ergodic theory) Gowers - 1998 Gowers - 2001

39. What about ? Szemerédi - 1969 + $1000 Szemerédi - 1974 Endre Szemerédi – (1940 - ) Furstenberg – 1979 (ergodic theory) Gowers - 1998 Gowers - 2001 The last result can be used to obtain the best currently available bound for the van der Waerden function W(k).

40. A consequence of van der Waerden’s theorem on arithmetic progressions is that for every n, there is a least number W(n) so that in any 2-coloring of [W(n)], there is always formed a monochromatic n-AP.

41. 2 2 2 ……… 2 n 2’s 2 2 2 2 2 W(n) < A consequence of van der Waerden’s theorem on arithmetic progressions is that for every n, there is a least number W(n) so that in any 2-coloring of [W(n)], there is always formed a monochromatic n-AP. Theorem(Gowers 2001) $1000

42. The best known lower bound is: (Berlekamp - 1968) Elwyn Berlekamp - (1940 - )

43. The only known values are: 2 3 n 4 5 6 35 9 3 1132 178 W(n) The best known lower bound is: (Berlekamp - 1968) Elwyn Berlekamp - (1940 - )

44. The only known values are: 2 3 n 4 5 6 35 9 3 1132 178 W(n) The best known lower bound is: (Berlekamp - 1968) Elwyn Berlekamp - (1940 - ) (Brave) Conjecture. $1000 For all n,

45. A new proof of van der Waerden for 3-AP’s

46. A new proof of van der Waerden for 3-AP’s Theorem: (RLG + J. Solymosi - 2006) There is a universal constant c such that if the points of G(N) are colored with at most c log log N colors, then there is always a monochromatic corner (x,y), (x+d,y), (x,y+d). “corner” N N G(N) = N by N triangular grid

47. Let L denote the set of points on the diagonal. 1 L 1 Sketch of proof: Suppose G(N) is r-colored. N N

48. Let L denote the set of points on the diagonal. 1 L 1 Sketch of proof: Suppose G(N) is r-colored. Let be points with the “most popular” color c . 1 N N

49. Let L denote the set of points on the diagonal. 1 Let be the points in the product below L . L 1 1 Sketch of proof: Suppose G(N) is r-colored. Let be points with the “most popular” color c . 1 N ……… ……… ……… N

50. Let L denote the set of points on the diagonal. 1 Let be the points in the product below L . L 1 1 Sketch of proof: Suppose G(N) is r-colored. Let be points with the “most popular” color c . 1 Thus, N ……… ……… ……… N