Transcript What is Ramsey Theory? What is Ramsey Theory? It might be described as the study of unavoidable regularity in large structures.

What is Ramsey Theory?
What is Ramsey Theory?
It might be described as the study of
unavoidable regularity in large structures.
What is Ramsey Theory?
It might be described as the study of
unavoidable regularity in large structures.
Complete disorder is impossible.
T. Motzkin
Ramsey’s Theorem (1930)
For any k < l and r, there exists R = R(k,l,r) so that
for any r-coloring of the k-element sets of an R-element
set, there is always some l-element set with all of its
k-element subsets having the same color.
Ramsey’s Theorem (1930)
For any k < l and r, there exists R = R(k,l,r) so that
for any r-coloring of the k-element sets of an R-element
set, there is always some l-element set with all of its
k-element subsets having the same color.
Frank Plumpton Ramsey
(1903-1930)
Euclidean Ramsey Theory
k
X  E - finite
Euclidean Ramsey Theory
k
X  E - finite
k
Cong(X) - family of all X  E which are congruent to X
(i.e., “copies” of X up to some Euclidean motion)
Euclidean Ramsey Theory
k
X  E - finite
k
Cong(X) - family of all X  E which are congruent to X
(i.e., “copies” of X up to some Euclidean motion)
X is said to be Ramsey if for all r there exists
N = N(X,r) such that for every partition E
N
= C1
we have X ‘  C i for some X ‘  Cong(X) and some i.
C2
.... Cr ,
Euclidean Ramsey Theory
k
X  E - finite
k
Cong(X) - family of all X  E which are congruent to X
(i.e., “copies” of X up to some Euclidean motion)
X is said to be Ramsey if for all r there exists
N = N(X,r) such that for every partition E
N
= C1
we have X ‘  C i for some X ‘  Cong(X) and some i.
E
N
r
X
C2
.... Cr ,
Compactness Principle
If E
N
r
such that Y
X
then there is a finite subset Y
r
X
EN
Compactness Principle
If E
N
r
X
then there is a finite subset Y
r
such that Y
X
Example
X=
1
|X| = 2
EN
Compactness Principle
If E
N
r
X
then there is a finite subset Y
r
such that Y
EN
X
Example
X=
1
|X| = 2
r
For a given r, take Yr  E to be the r+1 vertices of
r
a unit simplex in E .
Compactness Principle
If E
N
r
X
then there is a finite subset Y
r
such that Y
EN
X
Example
X=
1
|X| = 2
r
For a given r, take Yr  E to be the r+1 vertices of
r
a unit simplex in E .
Then Yr r
X.
n
n
Let Q denote the set of 2 vertices {(x1 ,..., xn ) : xk  0 or 1}
of the n-cube.
n
n
Let Q denote the set of 2 vertices {(x1 ,..., xn ) : xk  0 or 1}
of the n-cube. Then Q n is Ramsey.
n
n
Let Q denote the set of 2 vertices {(x1 ,..., xn ) : xk  0 or 1}
of the n-cube. Then Q n is Ramsey.
Theorem. For any k and r, there exists N = N(k,r) such that
N
any r-coloring of Q contains a monochromatic
k
2Q .
n
n
Let Q denote the set of 2 vertices {(x1 ,..., xn ) : xk  0 or 1}
of the n-cube. Then Q n is Ramsey.
Theorem. For any k and r, there exists N = N(k,r) such that
N
any r-coloring of Q contains a monochromatic
Idea of proof: (induction)
k=1
k
2Q .
Choose N(1,r) = r + 1
n
n
Let Q denote the set of 2 vertices {(x1 ,..., xn ) : xk  0 or 1}
of the n-cube. Then Q n is Ramsey.
Theorem. For any k and r, there exists N = N(k,r) such that
N
any r-coloring of Q contains a monochromatic
Idea of proof: (induction)
Consider the r + 1 points:
k=1
k
2Q .
Choose N(1,r) = r + 1
r+1
(1,0,0,......,0)
(0,1,0,......,0)
(0,0,1,......,0)
……
(0,0,0,......,1)
Since only r colors are used then some pair must have the same color, say
and
(……………,1,……………,0,……………)
(……………,0,……………,1,……………)
1
This is a monochromatic 2Q .
Since only r colors are used then some pair must have the same color, say
and
(……………,1,……………,0,……………)
(……………,0,……………,1,……………)
1
This is a monochromatic 2Q .
So far, so good!
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
Consider the N 2 N 1 points:
N2
N1
(1,0,0,…,0,1,0,0,…,0)
(1,0,0,…,0,0,1,0,…,0)
……
……
(1,0,0,…,0,0,0,0,…,1)
(0,1,0,…,0,1,0,0,…,0)
(0,1,0,…,0,0,1,0,…,0)
(0,1,0,…,0,0,0,0,…,1)
(0,0,1,…,0,1,0,0,…,0)
………
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
Consider the N 2 N 1 points:
N2
N1
(1,0,0,…,0,1,0,0,…,0)
(1,0,0,…,0,0,1,0,…,0)
……
……
(1,0,0,…,0,0,0,0,…,1)
(0,1,0,…,0,1,0,0,…,0)
(0,1,0,…,0,0,1,0,…,0)
(0,1,0,…,0,0,0,0,…,1)
(0,0,1,…,0,1,0,0,…,0)
………
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
Consider the N 2 N 1 points:
N2
N1
(1,0,0,…,0,1,0,0,…,0)
(1,0,0,…,0,0,1,0,…,0)
……
……
(1,0,0,…,0,0,0,0,…,1)
(0,1,0,…,0,1,0,0,…,0)
(0,1,0,…,0,0,1,0,…,0)
(0,1,0,…,0,0,0,0,…,1)
(0,0,1,…,0,1,0,0,…,0)
N2
(1,0,0,…,0)
1
1
.
1
N1
………
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
Consider the N 2 N 1 points:
N2
N1
(1,0,0,…,0,1,0,0,…,0)
(1,0,0,…,0,0,1,0,…,0)
……
……
(1,0,0,…,0,0,0,0,…,1)
(0,1,0,…,0,1,0,0,…,0)
(0,1,0,…,0,0,1,0,…,0)
(0,1,0,…,0,0,0,0,…,1)
(0,0,1,…,0,1,0,0,…,0)
N2
(1,0,0,…,0)
1
1
.
1
N1
………
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
Consider the N 2 N 1 points:
N2
N1
(1,0,0,…,0,1,0,0,…,0)
(1,0,0,…,0,0,1,0,…,0)
……
……
(1,0,0,…,0,0,0,0,…,1)
(0,1,0,…,0,1,0,0,…,0)
(0,1,0,…,0,0,1,0,…,0)
(0,1,0,…,0,0,0,0,…,1)
(0,0,1,…,0,1,0,0,…,0)
N2
1
1
(1,0,0,…,0)
.
1
N2
(0,1,0,…,0)
N1
1
1
N1
.
1
………
k=2
Choose N(2,r) = (r r+1 + 1) + (r + 1)
N2 + N1
=
Consider the N 2 N 1 points:
N2
N1
(1,0,0,…,0,1,0,0,…,0)
(1,0,0,…,0,0,1,0,…,0)
……
……
(1,0,0,…,0,0,0,0,…,1)
(0,1,0,…,0,1,0,0,…,0)
(0,1,0,…,0,0,1,0,…,0)
1
1
(1,0,0,…,0)
.
1
N2
(0,1,0,…,0)
N1
1
1
N1
.
1
………
………
(0,1,0,…,0,0,0,0,…,1)
(0,0,1,…,0,1,0,0,…,0)
N2
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
Since N 2 = r r+1 + 1 = r
N1
+ 1, some pair has the same coloring, say
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
Since N 2 = r r+1 + 1 = r
i
2
j
2
(………,1,………0,……)
i
N1
+ 1, some pair has the same coloring, say
j
1 1
1
0
0
1
(identical colorings)
(………,0,………1,……)
1
0
0
1
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
Since N 2 = r r+1 + 1 = r
i
2
j
2
i
N1
+ 1, some pair has the same coloring, say
j
1 1
(………,1,………0,……)
1
0
0
1
(………,0,………1,……)
1
0
0
1
monochromatic by choice of N1
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
Since N 2 = r r+1 + 1 = r
i
2
j
2
(………,1,………0,……)
i
N1
+ 1, some pair has the same coloring, say
j
1 1
1
0
0
1
Thus, all 4 are
monochromatic
(………,0,………1,……)
1
0
0
1
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
Since N 2 = r r+1 + 1 = r
i
2
j
2
(………,1,………0,……)
i
N1
+ 1, some pair has the same coloring, say
j
1 1
1
0
0
1
Thus, all 4 are
monochromatic
(………,0,………1,……)
1
0
0
1
2
These 4 points form a monochromatic 2 Q :
1
Since the N2 points represented by the
1
.
1
N 1‘s
can be r-colored in at most r N 1 ways, then the original
N2+ N
1 induces an r N 1 - coloring of QN 2.
r-coloring of Q
Since N 2 = r r+1 + 1 = r
i
2
j
2
(………,1,………0,……)
i
N1
+ 1, some pair has the same coloring, say
j
1 1
1
0
0
1
Thus, all 4 are
monochromatic
(………,0,………1,……)
1
0
0
1
2
These 4 points form a monochromatic 2 Q :
j
i2
j
i1
2
1
(……,1,………0,......,1,……,0,……)
(……,1,………0,......,0,……,1,……)
(……,0,………1,......,1,……,0,……)
(……,0,………1,......,0,……,1,……)
For k = 3, we can take N(3,r) = N3 + N2 + N 1
where
N2 N1
N3  1  r
1r
(1r1  r )(1 r)
, etc.
For k = 3, we can take N(3,r) = N3 + N2 + N 1
where
N2 N1
N3  1  r
1r
(1r1  r )(1 r)
Continuing this way, the theorem is proved.
, etc.
For k = 3, we can take N(3,r) = N3 + N2 + N 1
where
N2 N1
N3  1  r
1r
(1r1  r )(1 r)
, etc.
Continuing this way, the theorem is proved.
Note that by this technique, the bounds we get are rather large.
For k = 3, we can take N(3,r) = N3 + N2 + N 1
where
N2 N1
N3  1  r
1r
(1r1  r )(1 r)
, etc.
Continuing this way, the theorem is proved.
Note that by this technique, the bounds we get are rather large.
For example, it shows that N(4,2)
27
 2 + 13.
For k = 3, we can take N(3,r) = N3 + N2 + N 1
where
N2 N1
N3  1  r
1r
(1r1  r )(1 r)
, etc.
Continuing this way, the theorem is proved.
Note that by this technique, the bounds we get are rather large.
For example, it shows that N(4,2)
27
 2 + 13.
What is the true order of growth here?
With this technique, we can prove the:
Product Theorem. If X and Y are Ramsey then
the Cartesian product X x Y is also Ramsey.
With this technique, we can prove the:
Product Theorem. If X and Y are Ramsey then
the Cartesian product X x Y is also Ramsey.
Corollary: (Any subset of) the vertices of an n-dimensional
rectangular parallelepiped is Ramsey.
With this technique, we can prove the:
Product Theorem. If X and Y are Ramsey then
the Cartesian product X x Y is also Ramsey.
Corollary: (Any subset of) the vertices of an n-dimensional
rectangular parallelepiped is Ramsey.
For example, any acute triangle is Ramsey.
With this technique, we can prove the:
Product Theorem. If X and Y are Ramsey then
the Cartesian product X x Y is also Ramsey.
Corollary: (Any subset of) the vertices of an n-dimensional
rectangular parallelepiped is Ramsey.
For example, any acute triangle is Ramsey.
What about
?
How can we get obtuse Ramsey triangles?
How can we get obtuse Ramsey triangles?
Example.
Choose n = R(7, 9, r) and consider the set S of points x in
E
having all coordinates zero except for 7 coordinates which
have in order the values 1, 2, 3, 4, 3, 2, 1.
n
How can we get obtuse Ramsey triangles?
Example.
Choose n = R(7, 9, r) and consider the set S of points x in
E
having all coordinates zero except for 7 coordinates which
have in order the values 1, 2, 3, 4, 3, 2, 1.
x = (0 0 0 1 0 2 0 0 0 3 4 0 0 3 0 0 0 0 2 0 1 0 0 0 0)
n
How can we get obtuse Ramsey triangles?
Example.
Choose n = R(7, 9, r) and consider the set S of points x in
E
having all coordinates zero except for 7 coordinates which
have in order the values 1, 2, 3, 4, 3, 2, 1.
x = (0 0 0 1 0 2 0 0 0 3 4 0 0 3 0 0 0 0 2 0 1 0 0 0 0)
There are
( n7 )
such points in S.
n
How can we get obtuse Ramsey triangles?
Example.
Choose n = R(7, 9, r) and consider the set S of points x in
E
n
having all coordinates zero except for 7 coordinates which
have in order the values 1, 2, 3, 4, 3, 2, 1.
x = (0 0 0 1 0 2 0 0 0 3 4 0 0 3 0 0 0 0 2 0 1 0 0 0 0)
There are
( n7 )
such points in S.
Any r-coloring of S induces an r-coloring of the 7-sets of {1,2,……,n}
By the choice of n = R(7, 9, r), there exists some 9-set {i1 ,i2 ,...,i9 }
with all its 7-sets having the same color.
By the choice of n = R(7, 9, r), there exists some 9-set {i1 ,i2 ,...,i9 }
with all its 7-sets having the same color.
x = (……xi ……xi ……xi ……xi ……xi ……xi ……xi……xi……xi ……)
1
2
3
4
5
6
7
8
9
A = (…….1…….2…….3……..4…….3…….2…….1……..0…….0……)
B = (…….0…….1…….2……..3…….4…….3…….2……..1…….0……)
C = (…….0…….0…….1……..2…….3…….4…….3……..2…….1……)
By the choice of n = R(7, 9, r), there exists some 9-set {i1 ,i2 ,...,i9 }
with all its 7-sets having the same color.
x = (……xi ……xi ……xi ……xi ……xi ……xi ……xi……xi……xi ……)
1
2
3
4
5
6
7
8
9
A = (…….1…….2…….3……..4…….3…….2…….1……..0…….0……)
B = (…….0…….1…….2……..3…….4…….3…….2……..1…….0……)
C = (…….0…….0…….1……..2…….3…….4…….3……..2…….1……)
dist(A, B) = 8
dist(B, C) =
8
dist(A, C) = 26
By the choice of n = R(7, 9, r), there exists some 9-set {i1 ,i2 ,...,i9 }
with all its 7-sets having the same color.
x = (……xi ……xi ……xi ……xi ……xi ……xi ……xi……xi……xi ……)
1
2
3
4
5
6
7
8
9
A = (…….1…….2…….3……..4…….3…….2…….1……..0…….0……)
B = (…….0…….1…….2……..3…….4…….3…….2……..1…….0……)
C = (…….0…….0…….1……..2…….3…….4…….3……..2…….1……)
dist(A, B) = 8
dist(B, C) =
8
Thus, the ( 8 , 8 , 26 )-triangle is Ramsey.
dist(A, C) = 26
In general, this technique shows that the triangle with side lengths
2t ,
2t and
8t  6 is Ramsey.
In general, this technique shows that the triangle with side lengths
2t ,
2t and
8t  6 is Ramsey.
Note that the angle t between the short sides
t
A
B
8t - 6
C
o
180 as t
.
In general, this technique shows that the triangle with side lengths
2t ,
2t and
8t  6 is Ramsey.
Note that the angle t between the short sides
t
Form the product:
A
B
8t - 6
B’
C
C’
o
180 as t
.
In general, this technique shows that the triangle with side lengths
2t ,
2t and
8t  6 is Ramsey.
Note that the angle t between the short sides
t
Form the product:
A
B
8t - 6
o
180 as t
B’
C
C’
By the product theorem, triangle AB’C is also Ramsey.
.
In general, this technique shows that the triangle with side lengths
2t ,
2t and
8t  6 is Ramsey.
Note that the angle t between the short sides
t
Form the product:
A
B
8t - 6
o
180 as t
B’
C
C’
By the product theorem, triangle AB’C is also Ramsey.
Theorem (Frankl, Rödl) All triangles are Ramsey.
.
Theorem: (Frankl/Rödl – 1990)
For any (non-degenerate) simplex S
there is a c = c(S) so that
E
c log r
r
S
Ek,
Theorem: (Frankl/Rödl – 1990)
For any (non-degenerate) simplex S
there is a c = c(S) so that
E
What about
c log r
r
S
?
Ek,
Theorem (I. Kríz – 1991)
If X  E N has a transitive solvable group of isometries
then X is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive solvable group of isometries
then X is Ramsey.
Corollary. The set of vertices of any regular n-gon is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive solvable group of isometries
then X is Ramsey.
Corollary. The set of vertices of any regular n-gon is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive group of isometries which has
a solvable subgroup with at most 2 orbits then X is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive solvable group of isometries
then X is Ramsey.
Corollary. The set of vertices of any regular n-gon is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive group of isometries which has
a solvable subgroup with at most 2 orbits then X is Ramsey.
Corollary. The set of vertices of any Platonic solid is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive solvable group of isometries
then X is Ramsey.
Corollary. The set of vertices of any regular n-gon is Ramsey.
Theorem (I. Kríz – 1991)
If X  E N has a transitive group of isometries which has
a solvable subgroup with at most 2 orbits then X is Ramsey.
Corollary. The set of vertices of any Platonic solid is Ramsey.
Are there any non-Ramsey sets??
Proof that
1
1
is not Ramsey.
.
Proof that
1
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
.
Proof that
1
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
(alternating spherical shells about O with decreasing thickness)
.
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
1
z
c
a
O
.
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
1
z
Then
a2  b2  1  2bcos 
c
a
O
.
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
1
z
Then
a2  b2  1  2bcos 
c2  b2  1  2bcos 
c
a
O
.
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
a
1
c
Then
a2  b2  1  2bcos 
z
c2  b2  1  2bcos 
Thus,
a2  c2  2b2  2
O
.
(*)
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
a
O
1
c
z
Then
a2  b2  1  2bcos 
c2  b2  1  2bcos 
a  c  2b  2
Thus,
_ _
_
If x, y and z have color d, then
a2  4ka  d   a , 0   a  1
2
2
2
.
(*)
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
a
O
1
c
z
Then
a2  b2  1  2bcos 
c2  b2  1  2bcos 
a  c  2b  2
Thus,
_ _
_
If x, y and z have color d, then
a2  4ka  d   a , 0   a  1
b2  4kb  d  b , 0  b  1
c2  4kc  d  c , 0  c  1
2
2
2
.
(*)
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
a
O
By (*),
1
c
z
Then
a2  b2  1  2bcos 
c2  b2  1  2bcos 
a  c  2b  2
Thus,
_ _
_
If x, y and z have color d, then
a2  4ka  d   a , 0   a  1
b2  4kb  d  b , 0  b  1
c2  4kc  d  c , 0  c  1
2
2
2
4ka  4kc  2d  a  c  8kb  2d  2b  2
.
(*)
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
a
O
By (*),
1
c
z
Then
a2  b2  1  2bcos 
c2  b2  1  2bcos 
a  c  2b  2
Thus,
_ _
_
If x, y and z have color d, then
a2  4ka  d   a , 0   a  1
b2  4kb  d  b , 0  b  1
c2  4kc  d  c , 0  c  1
2
2
2
4ka  4kc  2d  a  c  8kb  2d  2b  2
i.e.,
4M  2  2b  a  c
.
(*)
1
Proof that
1
is not Ramsey.
N
4-color each x  E according to |- x x -| (mod 4).
1
x
y
 b
a
O
By (*),
1
c
Then
a2  b2  1  2bcos 
z
c2  b2  1  2bcos 
a  c  2b  2
Thus,
_ _
_
If x, y and z have color d, then
a2  4ka  d   a , 0   a  1
b2  4kb  d  b , 0  b  1
c2  4kc  d  c , 0  c  1
2
2
2
4ka  4kc  2d  a  c  8kb  2d  2b  2
i.e.,
4M  2  2b  a  c
which is impossible since
2  2b  a  c  2 .
(*)
Call X spherical if X is a subset of some sphere Sd () in E k
Call X spherical if X is a subset of some sphere Sd () in E k
dimension
radius
Call X spherical if X is a subset of some sphere Sd () in E k
,,
Theorem (Erdos, Graham, Montgomery, Rothschild, Spencer, Straus)
X is Ramsey

X is spherical.
Call X spherical if X is a subset of some sphere Sd () in E k
,,
Theorem (Erdos, Graham, Montgomery, Rothschild, Spencer, Straus)
X is Ramsey

X is spherical.
Corollary.
X=
a
b
(collinear) is not Ramsey.
Call X spherical if X is a subset of some sphere Sd () in E k
,,
Theorem (Erdos, Graham, Montgomery, Rothschild, Spencer, Straus)
X is Ramsey

X is spherical.
Corollary.
X=
In fact, E
a
N
b
16
X
(collinear) is not Ramsey.
for any N.
Call X spherical if X is a subset of some sphere Sd () in E k
,,
Theorem (Erdos, Graham, Montgomery, Rothschild, Spencer, Straus)
X is Ramsey

X is spherical.
Corollary.
X=
In fact, E
a
N
b
16
X
Is 16 best possible??
(collinear) is not Ramsey.
for any N.
Definition: X is called sphere-Ramsey if for all r, there exist
N = N(X,r) and  =  (X,r) such that for all partitions
Sd () = C1
C2
.... Cr , some Ci contains a copy of X.
Definition: X is called sphere-Ramsey if for all r, there exist
N = N(X,r) and  =  (X,r) such that for all partitions
Sd () = C1
C2
.... Cr , some Ci contains a copy of X.
Note: sphere-Ramsey

Ramsey

spherical
Theorem (Matoušek/Rödl)
d
If X  S (1) is a simplex then for all r and all  > 0,
there exists N = N (X, r,  ) such that
Sd (1  )
r
X
Theorem (Matoušek/Rödl)
d
If X  S (1) is a simplex then for all r and all  > 0,
there exists N = N (X, r,  ) such that
Sd (1  )
r
Thus, X is sphere-Ramsey.
X
Theorem (Matoušek/Rödl)
d
If X  S (1) is a simplex then for all r and all  > 0,
there exists N = N (X, r,  ) such that
Sd (1  )
r
Thus, X is sphere-Ramsey.
Is the  really needed?
X
Theorem (Matoušek/Rödl)
d
If X  S (1) is a simplex then for all r and all  > 0,
there exists N = N (X, r,  ) such that
Sd (1  )
r
X
Thus, X is sphere-Ramsey.
Is the  really needed?
Yes !
Theorem (RLG)
Suppose X  {x1,...,xk }  Sd (1) is unit-sphere-Ramsey
(i.e., SN (1)
r
X, N = N(X,r) )
Theorem (RLG)
Suppose X  {x1,...,xk }  Sd (1) is unit-sphere-Ramsey
(i.e., SN (1)
r
X, N = N(X,r) )
Then for any linear dependence
 cx  0,
iI
i i
there must exist a nonempty set J  I with
c
jJ
j
 0.
Theorem (RLG)
Suppose X  {x1,...,xk }  Sd (1) is unit-sphere-Ramsey
(i.e., SN (1)
r
X, N = N(X,r) )
Then for any linear dependence
 cx  0,
iI
i i
there must exist a nonempty set J  I with
c
jJ
Corollary. If X above has 0
(since 0 
 cx
iI
i i
j
 0.
conv(X) then X is not unit-sphere-Ramsey.
with all ci > 0 ).
n
Suppose that we fix the dimension of the space E.
What is true in this case?
n
Suppose that we fix the dimension of the space E.
What is true in this case?
The simplest set:
1
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
2
(E 22) 2??
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
2
(E 22) 2??
4  (E2 )  7
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
2
(E 22) 2??
Edward Nelson
4  (E2 )  7
(1950)
Mosers’ graph M
Mosers’ graph M
(E )  (M)  4
2
(E2 )  7
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
33 3
What about E ?
2
(E 22) 2??
4  (E2 )  7
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
33 3
What about E ?
2
(E 22) 2??
4  (E2 )  7
6  (E 333)  15
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
2
(E 22) 2??
4  (E2 )  7
6  (E 333)  15
Nechustan (2000)
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
2
(E 22) 2??
4  (E2 )  7
6  (E 333)  15
Nechustan (2000)
Radoicić/Tóth (2002)
22 2 2

(
E
), the chromatic number of E 2, to be
Define
the least r such for some r-coloring E2 = C1
C2
.... Cr ,
no Ci contains 2 points at a distance of 1 from each other.
In other words, no unit distance occurs monochromatically
What is the value of
For
2
(E 22) 2??
4  (E2 )  7
En it is known that:
(1  o(1))( 6 )n (En )  (3  o(1))n
5
Theorem (O’Donnell – 2000)
For every g, there is a 4-chromatic unit distance graph G
in
E2
having girth greater than g.
Theorem (O’Donnell – 2000)
For every g, there is a 4-chromatic unit distance graph G
in
E2
having girth greater than g.
This is perhaps evidence supporting the conjecture that:
?
(E )  5
2
Theorem (O’Donnell – 2000)
For every g, there is a 4-chromatic unit distance graph G
in
E2
having girth greater than g.
This is perhaps evidence supporting the conjecture that:
?
(E )  5
2
Problem: ($1000) Determine the value of
(E2 ) .
The beginnings
(E.Klein)
Any set X of 5 points in the plane in general position must
contain the vertices of a convex 4-gon.
For each n, let f(n) denote the least integer so that any set X
of f(n) points in the plane in general position must contain the
vertices of a convex n-gon.
For each n, let f(n) denote the least integer so that any set X
of f(n) points in the plane in general position must contain the
vertices of a convex n-gon.
Does f(n) always exist?
For each n, let f(n) denote the least integer so that any set X
of f(n) points in the plane in general position must contain the
vertices of a convex n-gon.
Does f(n) always exist?
If so, determine or estimate it.
,,
Erdos and Szekeres showed that f(n) always exists and, in fact,
2
n 2
 2n  4 
 1  f(n)   n  2   1


,,
Erdos and Szekeres showed that f(n) always exists and, in fact,
2
n 2
 2n  4 
 1  f(n)   n  2   1


They gave several proofs that f(n) exists, one of which
used their independent discovery of Ramsey’s Theorem.
2
n 2
 2n  4 
 1  f(n)   n  2   1


2
n 2
 2n  4 
 1  f(n)   n  2   1


 2n  4 
f(n)   n  2 


Chung/Graham (1997)
2
n 2
 2n  4 
 1  f(n)   n  2   1


 2n  4 
f(n)   n  2 


Chung/Graham (1997)
 2n  4 
f(n)   n  2   2n  7


Kleitman/Pachter (1997)
2
n 2
 2n  4 
 1  f(n)   n  2   1


 2n  4 
f(n)   n  2 


Chung/Graham (1997)
 2n  4 
f(n)   n  2   2n  7


 2n  5 
f(n)   n  2   2


Kleitman/Pachter (1997)
G. Tóth/Valtr (1997)
Conjecture ($1000)
f(n)  2n2  1, for n  2
More beginnings
van der Waerden’s Theorem (1927)
In any partition of
N = {1,2,3,……} in finitely many
classes C1  C2  ...  Cr , some Ci must contain
k-term arithmetic progressions for all k.
More beginnings
van der Waerden’s Theorem (1927)
In any partition of
N = {1,2,3,……} in finitely many
classes C1  C2  ...  Cr , some Ci must contain
k-term arithmetic progressions for all k.
B. L. van der Waerden
(1903-1996)
More beginnings
van der Waerden’s Theorem (1927)
In any partition of
N = {1,2,3,……} in finitely many
classes C1  C2  ...  Cr , some Ci must contain
k-term arithmetic progressions for all k.
k-AP
More beginnings
van der Waerden’s Theorem (1927)
In any partition of
N = {1,2,3,……} in finitely many
classes C1  C2  ...  Cr , some Ci must contain
k-term arithmetic progressions for all k.
k-AP
Erdös and Turán ask in 1936 which Ci has k-AP’s ?
They conjectured that if Ci is “dense enough” then
this should imply that Ci has k-AP’s.
They conjectured that if Ci is “dense enough” then
this should imply that Ci has k-AP’s.
Define rk(n) to be the least integer such that any set
X
 {1,2,…,n} with |X|  rk(n) must contain a k-AP.
They conjectured that if Ci is “dense enough” then
this should imply that Ci has k-AP’s.
Define rk(n) to be the least integer such that any set
X
 {1,2,…,n} with |X|  rk(n) must contain a k-AP.
,,
Erdos and Turán conjectured that rk(n) = o(n).
Progress was slow

r3 (n)  n exp c log n

Behrend (1946)
Progress was slow

r3 (n)  n exp c log n
r3 (n)  O( n
(loglog n)

c)
Behrend (1946)
Roth (1954)
Progress was slow

r3 (n)  n exp c log n
r3 (n)  O( n
r4 (n)  o(n)
(loglog n)

c)
Behrend (1946)
Roth (1954)
Szemerédi (1969)
Progress was slow

r3 (n)  n exp c log n
r3 (n)  O( n
r4 (n)  o(n)
(loglog n)

c)
Behrend (1946)
Roth (1954)
Szemerédi (1969)
rk (n)  o(n) for all k
Szemerédi (1974)
Progress was slow

r3 (n)  n exp c log n
r3 (n)  O( n
r4 (n)  o(n)
(loglog n)

c)
Behrend (1946)
Roth (1954)
Szemerédi (1969)
rk (n)  o(n) for all k
Szemerédi (1974)
($1000 and the regularity lemma)
MATHEMATICS:
Erdös's Hard-to-Win Prizes Still Draw Bounty Hunters
Charles Seife – Science Magazine (April, 2002)
Half a decade after his death, a colorful mathematician continues to spur his
colleagues with challenges--and checks
It takes more than death to keep a good mathematician down. Among many other things,
Paul Erdös proved that. In life, the world's premier number theorist supported himself by
wandering from colleague to colleague, freeloading off friends and strangers, and, in
return, sharing his vast mathematical insight with all comers. His 6-decade frenzy of
near-nonstop work resulted in more than 1500 papers that link him to almost every
academic mathematician in the world. Erdös's death in 1996 has slowed, but not stopped,
his publication rate: Over the past 5 years, journals have published some 62 new papers
bearing his name. And he is still guiding the research of mathematicians with the problems
he left behind.
Progress was slow

r3 (n)  n exp c log n
r3 (n)  O( n
r4 (n)  o(n)
(loglog n)

c)
Behrend (1946)
Roth (1954)
Szemerédi (1969)
rk (n)  o(n) for all k
Szemerédi (1974)


n
r3 (n)  O 

 (log n) 13 


Heath-Brown (1987), Szemerédi (1990)
Progress was now starting to speed up

r3 (n)  n exp c log n
r3 (n)  O( n
(loglog n)
r4 (n)  o(n)

Behrend (1946)
c)
Roth (1954)
Szemerédi (1969)
rk (n)  o(n) for all k
Szemerédi (1974)


n
r3 (n)  O 

 (log n) 13 


Heath-Brown (1987), Szemerédi (1990)
r4 (n)  O( n
(loglog n)
c)
Gowers (1998)
Progress is now accelerating

r3 (n)  n exp c log n
r3 (n)  O( n
(loglog n)
r4 (n)  o(n)

Behrend (1946)
c)
Roth (1954)
Szemerédi (1969)
rk (n)  o(n) for all k
Szemerédi (1974)


n
r3 (n)  O 

 (log n) 13 


Heath-Brown (1987), Szemerédi (1990)
r4 (n)  O( n
(loglog n)
c)
Gowers (1998)
Progress is now accelerating


r3 (n)  n exp c log n
r3 (n)  O( n
(loglog n)
r4 (n)  o(n)
Behrend (1946)
c)
Roth (1954)
Szemerédi (1969)
rk (n)  o(n) for all k
Szemerédi (1974)


n
r3 (n)  O 

 (log n) 13 


Heath-Brown (1987), Szemerédi (1990)
r4 (n)  O( n
rk (n)  O( n
(loglog n)
c)
ck )
(loglog n)
Gowers (1998)
Gowers (2000)
Define W(n) to be the least integer W (by van der
Waerden) so that every 2-coloring of {1,2,…,W} has
an n-AP.
Define W(n) to be the least integer W (by van der
Waerden) so that every 2-coloring of {1,2,…,W} has
an n-AP.
Corollary (Gowers 2000)
2n  9
2
22
W(n)  2
, for all n.
Define W(n) to be the least integer W (by van der
Waerden) so that every 2-coloring of {1,2,…,W} has
an n-AP.
Corollary (Gowers 2000)
($1000)
2n  9
2
22
W(n)  2
, for all n.
Define W(n) to be the least integer W (by van der
Waerden) so that every 2-coloring of {1,2,…,W} has
an n-AP.
Corollary (Gowers 2000)
2n  9
2
22
W(n)  2
Conjecture ($1000): W(n)  2
, for all n.
n2
for all n.
Define W(n) to be the least integer W (by van der
Waerden) so that every 2-coloring of {1,2,…,W} has
an n-AP.
Corollary (Gowers 2000)
2n  9
2
22
W(n)  2
Conjecture ($1000): W(n)  2
, for all n.
n2
for all n.
Best current lower bound is W(n+1)  n .2n, n prime (Berlekamp 1968)
2
What can be true for partitions of E if we allow
an arbitrary finite number of colors?
2
What can be true for partitions of E if we allow
an arbitrary finite number of colors?
Theorem. (RLG) For every r, there exists a least integer T(r)
so that for any partition of Z2 = C1
C2
.... Cr ,
some Ci contains the vertices of a triangle of area exactly T(r).
2
What can be true for partitions of E if we allow
an arbitrary finite number of colors?
Theorem. (RLG) For every r, there exists a least integer T(r)
so that for any partition of Z2 = C1
C2
.... Cr ,
some Ci contains the vertices of a triangle of area exactly T(r).
How large is T(r)?
2
What can be true for partitions of E if we allow
an arbitrary finite number of colors?
Theorem. (RLG) For every r, there exists a least integer T(r)
so that for any partition of Z2 = C1
C2
.... Cr ,
some Ci contains the vertices of a triangle of area exactly T(r).
How large is T(r)?
It can be shown that T(r) > (21) l.c.m (2,3,…,r) = e
(1 o(1))r
.
2
What can be true for partitions of E if we allow
an arbitrary finite number of colors?
Theorem. (RLG) For every r, there exists a least integer T(r)
so that for any partition of Z2 = C1
C2
.... Cr ,
some Ci contains the vertices of a triangle of area exactly T(r).
How large is T(r)?
It can be shown that T(r) > (21) l.c.m (2,3,…,r) = e
(1 o(1))r
The best known upper bound grows much faster than
the (infamous) van der Waerden function W.
.
For example, let W(k,r) denote the least value W so that in
any r-coloring of the first W integers, there is always formed a
monochromatic k-term arithmetic progression.
Then T(3) < 725760 .1725761
For example, let W(k,r) denote the least value W so that in
any r-coloring of the first W integers, there is always formed a
monochromatic k-term arithmetic progression.
Then T(3) < 725760 .1725761!
For example, let W(k,r) denote the least value W so that in
any r-coloring of the first W integers, there is always formed a
monochromatic k-term arithmetic progression.
Then T(3) < 725760 .1725761! W(725761!+1,3)
For example, let W(k,r) denote the least value W so that in
any r-coloring of the first W integers, there is always formed a
monochromatic k-term arithmetic progression.
Then T(3) < 725760 .1725761! W(725761!+1,3)!
For example, let W(k,r) denote the least value W so that in
any r-coloring of the first W integers, there is always formed a
monochromatic k-term arithmetic progression.
Then T(3) < 725760 .1725761! W(725761!+1,3)!
Actually, T(3) = 3.
For example, let W(k,r) denote the least value W so that in
any r-coloring of the first W integers, there is always formed a
monochromatic k-term arithmetic progression.
Then T(3) < 725760 .1725761! W(725761!+1,3)!
Actually, T(3) = 3.
What is the truth here??
What if you allow infinitely many colors?
What if you allow infinitely many colors?
Theorem (Kunen)
Assuming the Continuum Hypothesis, it is possible to partition E
into countably many sets, none of which contains the vertices
of a triangle with rational area.
2
What if you allow infinitely many colors?
Theorem (Kunen)
Assuming the Continuum Hypothesis, it is possible to partition E
into countably many sets, none of which contains the vertices
of a triangle with rational area.
,,
Theorem (Erdos/Komjáth)
2
The existence of a partition of E into countably many sets,
none of which contains the vertices of a right triangle is
equivalent to the Continuum Hypothesis.
2
Edge-Ramsey Configurations
A finite configuration L of line segments in
E k is said to
be edge-Ramsey if for any r there is an N = N(L,r) so that
in any r-coloring of the line segments in E N
, there is always
a monochromatic copy of L.
Edge-Ramsey Configurations
A finite configuration L of line segments in
E k is said to
be edge-Ramsey if for any r there is an N = N(L,r) so that
in any r-coloring of the line segments in E N
, there is always
a monochromatic copy of L.
What do we know about edge-Ramsey configurations?
Theorem (EGMRSS)
If L is edge-Ramsey then all the edges of L must have the same length.
Theorem (EGMRSS)
If L is edge-Ramsey then all the edges of L must have the same length.
Theorem (RLG)
If L is edge-Ramsey then the endpoints of the edges of E
must lie on two spheres.
Theorem (EGMRSS)
If L is edge-Ramsey then all the edges of L must have the same length.
Theorem (RLG)
If L is edge-Ramsey then the endpoints of the edges of E
must lie on two spheres.
Theorem (RLG)
If the endpoints of the edges of L do not lie on a sphere and
the graph formed by L is not bipartite then L is not edge-Ramsey.
Theorem (Cantwell)
The edge set of an n-cube is edge-Ramsey.
Theorem (Cantwell)
The edge set of an n-cube is edge-Ramsey.
Theorem (Cantwell)
The edge set of a regular n-gon is not edge-Ramsey if n = 5 or n > 6.
Theorem (Cantwell)
The edge set of an n-cube is edge-Ramsey.
Theorem (Cantwell)
The edge set of a regular n-gon is not edge-Ramsey if n = 5 or n > 6.
Question: Is the edge set of a regular hexagon edge-Ramsey?
Theorem (Cantwell)
The edge set of an n-cube is edge-Ramsey.
Theorem (Cantwell)
The edge set of a regular n-gon is not edge-Ramsey if n = 5 or n > 6.
Question: Is the edge set of a regular hexagon edge-Ramsey?
(Big) Problem: Characterize edge-Ramsey configurations.
Theorem (Cantwell)
The edge set of an n-cube is edge-Ramsey.
Theorem (Cantwell)
The edge set of a regular n-gon is not edge-Ramsey if n = 5 or n > 6.
Question: Is the edge set of a regular hexagon edge-Ramsey?
(Big) Problem: Characterize edge-Ramsey configurations.
There is currently no plausible conjecture.
We know:
sphere-Ramsey

Ramsey

spherical

rectangular
We know:
sphere-Ramsey

Ramsey

What about the other direction?
spherical

rectangular
We know:
sphere-Ramsey

Ramsey

spherical
What about the other direction?
sphere-Ramsey
?

Ramsey

rectangular
We know:
sphere-Ramsey

Ramsey

spherical

rectangular
What about the other direction?
sphere-Ramsey
?

Ramsey
?

spherical
We know:
sphere-Ramsey

Ramsey

spherical

rectangular
What about the other direction?
sphere-Ramsey
?
?
Ramsey 

$1000
spherical
We know:
sphere-Ramsey

Ramsey

spherical

rectangular
What about the other direction?
sphere-Ramsey
?
?
Ramsey 

$1000
$1000
spherical
We know:
sphere-Ramsey

Ramsey

spherical

rectangular
What about the other direction?
sphere-Ramsey
?
?
Ramsey 

$1000
$1000
I’ll close with some easier(?) problems:
spherical
Question: What are the unit-sphere-Ramsey configurations?
Question: What are the unit-sphere-Ramsey configurations?
Conjecture ($50)
For any triangle T, there is a 3-coloring of E
with no monochromatic copy of T.
2
Question: What are the unit-sphere-Ramsey configurations?
Conjecture ($50)
For any triangle T, there is a 3-coloring of E
2
with no monochromatic copy of T.
Conjecture ($100):
2 2
Every 2-coloring of E22232
contains
a monochromatic copy of
every triangle, except possibly for a single equilateral triangle.
Question: What are the unit-sphere-Ramsey configurations?
Conjecture ($50)
For any triangle T, there is a 3-coloring of E
2
with no monochromatic copy of T.
Conjecture ($100):
2 2
Every 2-coloring of E22232
contains
a monochromatic copy of
every triangle, except possibly for a single equilateral triangle.
Conjecture ($100)
Any 4-point subset of a circle is Ramsey.
Question: What are the unit-sphere-Ramsey configurations?
Conjecture ($50)
For any triangle T, there is a 3-coloring of E
2
with no monochromatic copy of T.
Conjecture ($100):
2 2
Every 2-coloring of E22232
contains
a monochromatic copy of
every triangle, except possibly for a single equilateral triangle.
Conjecture ($100)
Any 4-point subset of a circle is Ramsey.
Conjecture ($1000)
Every spherical set is Ramsey.