Thresholds for Ackermannian Ramsey Numbers. Authors: Menachem Kojman Gyesik Lee Eran Omri Andreas Weiermann. Notation …. n = {1..n} means: for every coloring C of the edges of the complete graph K n , there is a complete Q monochromatic sub-graph of size k .
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Authors: Menachem Kojman
Gyesik Lee
Eran Omri
Andreas Weiermann
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
Let us prove:
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.
Find amin-homogeneouscomplete sub-graph of size2k.
if for every v in V, all edges (v,u), for
u > v, are assigned one color.
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
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)
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)
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.
Denote Rc(k) :=The minimumnto satisfy:
To sum up:
R2(k) – Exponential in k
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
A coloring C is g-regressive if for every (m,n)
C(m,n) ≤ g(min(m,n)) = g(m)
Observe that
Is true for any g: N N which can be established by means,similar to the regular Ramsey proof and compactness.
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
That for g = ID:
Rg(k) is Ackermannian in terms of k.
Namely, DOMINATES every primitive recursive function.
Rg(k) < gk (primitive recursive.)
Rg(k) is ackermannian.
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
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.
Basic Pointers:
Set c = gB(n)
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).
a bad coloring of an Akermannianly large n
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}
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
Given a monotonically increasing function 4g2and a natural number k >2 with
we define a coloring
That is:
x1/j - Ackermannian
I will insert a drawing
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.
Input:Tuples of natural numbers
Output:A natural number
Basic primitive recursive functions:
More complex primitive recursive functions are obtained by :
General Definition – (fg) Hierarchy
Where f0(n) = n and fj+1(n) = f(fj(n))
Ackermann’s Function
Denote: Ack(n) = An(n)
Ackermann’s Function
(Erdös & Rado – 1950)
Definition…
Examples…
(Erdös & Rado – 1950)
2. On the other handthere exist t,l such that:
Now, since
We have
And thus:
k
i
3
2
1
µµ+1 µ+2 …
There exists no:
Which is min-homogeneous for Cg... But, for every (m,n) in the interval,
To do that we used a general, well known, coloring method…
Imagine yourself in a billiard hall…
A tournament is being organized…
6
Ramsey Number for 3 is 6.
Pool=
Snooker =
General Definition – (fg) Hierarchy
Where f0(n) = n and fj+1(n) = f(fj(n))
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.