Thresholds for Ackermannian Ramsey Numbers

1. Thresholds for Ackermannian Ramsey Numbers Authors: Menachem Kojman Gyesik Lee Eran Omri Andreas Weiermann

2. Notation… • n = {1..n} • means: for every coloring C of the edges of the complete graph Kn, there is a complete Q monochromatic sub-graph of size k. n–Size of complete graph over which we color edges. k–Size of homogeneous sub-graph. c–Number of colors. 2–Size of tuples we color – pairs (edges). 2 Q is homogeneous for C Example

3. Example… Let us prove: • means: for every coloring C of the edges of the complete graph Kn, there is a complete Q monochromatic sub-graph of size k. n = 22k-1–Size of complete graph over which we color edges. k–Size of homogeneous sub-graph. c = 2–Number of colors.

4. Proof… • Let us, from now on, assume the vertex set of every graph we consider is some initial segment of the natural numbers. • First step: Find amin-homogeneouscomplete sub-graph of size2k.

5. Definition:A complete graph G = (E,V), with the natural ordering on V, is min-homogeneous for a coloring if for every v in V, all edges (v,u), for u > v, are assigned one color.

6. …Proof… Let G = (V,E) be the complete graph with V= 22k-1: x2 x3 x4 x5 x6 x1 … x22k-1 … And let C be a coloring of the vertices of G with 2colors (Red, Blue). C(x1,x2) = redC(x2,x6) = blue

7. …Proof… Markxiwith the colorC(x1, xi) x2 x3 x4 x5 x1 x6 … … x22k-1 … There is a monochromatic complete sub-graph of{x2,x3,…} of size 22k-2. (Sayred)

8. …Proof… Mark xi with the colorC(x2, xi) x2 x3 x6 x1 … There is a monochromatic complete sub-graph of{x3,x6,…} of size 22k-3. (Sayblue)

9. …Proof… Now, we have amin-homogeneoussequence {xi1, xi2, xi3, …, xi2k} xi2 xi3 xi4 xi5 xi1 xi6 xi2k … Mark xiawith C(xia, xib) for allb > a. There is a monochromatic subset of {xi1, xi2, xi3, … xi2k} of size k. • Second (and final) step: • Find a k-sizedhomogeneouscomplete sub-graph

10. Ramsey Numbers. • We Showed: • On the other Hand: -Using the Probabilistic Method, we can show: Denote Rc(k) :=The minimumnto satisfy: To sum up: R2(k) – Exponential in k

11. More importantly: • Explanation: - For a k-sized sequence iterate step #1 (repeated division) k times (namely, divide by at most c at each iteration)

12. Primitive Recursive Functions and Ackermann’s Function A function that can be implemented using only for-loops is calledprimitive recursive. Ackermann’s function– A simple example of a well defined total function that is computable but not primitive recursive

13. Regressive Ramsey…. • g-regressive colorings: A coloring C is g-regressive if for every (m,n) C(m,n) ≤ g(min(m,n)) = g(m) • Can we still demand homogeneity?? • Not necessarily!!! ( e.g C(m,n) = Id(m) ) Observe that Is true for any g: N  N which can be established by means,similar to the regular Ramsey proof and compactness.

14. The g-regressive Ramsey Number Denote Rg(k) :=The minimumnto satisfy: We have seen so far: For a constant function,g(x) = c Rg(k) ≤ ck Since

15. On the other hand it was known… That for g = ID: Rg(k) is Ackermannian in terms of k. Namely, DOMINATES every primitive recursive function.

16. The Problem • constant g Rg(k) < gk (primitive recursive.) • Threshold g Rg(k) is ackermannian. • g = Id

17. The Results If g(n) is ‘fast’ to go below n1/kthen, Rg(k) is primitive recursive If g(n)≤n1/kthen, x1/j - Ackermannian I will insert a drawing

18. The Results Suppose B : N  N is positive, unbounded and non-decreasing. Let gB(n) = n1/B−1(n). Where B-1(n) = min{t : B(t) ≥ n}. Then, RgB(k) is Ackermannian iff B is Ackermannian.

19. Min-homogeneity – Lower Threshold Basic Pointers: • Assume more colors. Set c = gB(n) • Use repeated division to show min B(k) ≥ ck (k)gB Suppose B : N  N is positive, unbounded and non-decreasing. Let gB(n) = n1/B−1(n). Where B-1(n) = min{t : B(t) ≥ n}. Then, for every natural number k, it holds that RgB(k)≤ B(k).

20. Min-homogeneity – Upper Threshold • We show: • To prove this, we present, given k, a bad coloring of an Akermannianly large n

21. The Bad Coloring C(m,n) = <I,D> I Largest i s.t m,n are not in same segment. D  Distance between m’s segment and n’s. (fg)i(µ) k (fg)3(2)(µ) i (fg)3(µ) 3 2 1 µµ+1 µ+2 … {(fg)i}

22. The Bad Coloring C(m,n) = <I,D> = <2,1> I Largest i s.t m,n are not in same segment. D  Distance between m’s segment and n’s. k i 3 2 1 µµ+1 µ+2 … m = µ+8 n = µ+29

23. The Coloring – Formal Definition Given a monotonically increasing function 4g2and a natural number k >2 with we define a coloring That is: • 4g2-regressive on the interval • Has no min-homogeneous set of size k+1 within that interval.

24. Definitions…

25. The Coloring… • So, why is it: • 4g2-regressive? • Avoiding a min-homogeneous set?

26. The Results - Surfing the waves x1/j - Ackermannian I will insert a drawing

27. Besides The Asymptotic bounds,We can also establish:

28. omrier@cs.bgu.ac.il

29. In 1985 Kanamori & McAloon Had used means of Model Theory to show that the bound of : eventually DOMINATES every primitive recursive function. In 1991 Prömel, Thumser & Voigt and independently in 1999 Kojman & Shelah have presented two simple combinatorial proofs to this fact.

30. Primitive Recursive Functions Input:Tuples of natural numbers Output:A natural number Basic primitive recursive functions: • The constant function 0 • The successor function S • The projection functions Pin(x1, x2,…, xn) = xi

31. Primitive Recursive Functions More complex primitive recursive functions are obtained by : • Composition:h(x0,...,xl-1) = f(g0(x0,...,xl-1),...,gk-1(x0,...,xl-1)) • Primitive recursion:h(0,x0,...,xk-1) = f(x0,...,xk-1) h(S(n),x0,...,xk-1) = g(h(n,x0,...,xk-1),n,x0,...,xk-1)

32. Given a function g : N N, denote General Definition – (fg) Hierarchy Where f0(n) = n and fj+1(n) = f(fj(n))

33. Let g = Id. Now Ackermann’s Function Denote: Ack(n) = An(n)

34. Examples: Ackermann’s Function

35. Infinite Canonical Ramsey theorem (Erdös & Rado – 1950) Definition… Examples… • Finite Canonical Ramsey theorem (Erdös & Rado – 1950)

38. 1. clearly: 2. On the other handthere exist t,l such that: Now, since We have And thus:

39. The Bad Coloring k i 3 2 1 µµ+1 µ+2 …

40. There exists no: Which is min-homogeneous for Cg... But, for every (m,n) in the interval,

41. Homogeneity – Lower Bound • We show: • To prove that we used:

42. Homogeneity – Upper Bound • We showed: To do that we used a general, well known, coloring method…

43. The s-basis coloring

44. Example… • The rules: • Any two players may choose to play Pool or Snooker. (Coloring pairs with two colors) • A tournament can take place either in Snooker or in Pool. All the couples must choose the same. (Homogeneous set) • Three players minimum. (Size of subset) Imagine yourself in a billiard hall… A tournament is being organized…

45. How many players will ensure a Tournament?? 6 Ramsey Number for 3 is 6. Pool = Snooker =

46. Given a function g : N N, denote General Definition – (fg) Hierarchy Where f0(n) = n and fj+1(n) = f(fj(n))

47. Suppose g : N  N is nondecreasing and unbounded. Then, Rg(k) is bounded by some primitive recursive function in k iff for every t > 0 there is some M(t) s.t for all n ≥ M(t) it holds that g(n) < n1/t and M(t) is primitive recursive in t.