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Department of Mathematics, Nanjing University Email: [email protected] 7 April, 2021 Yaojun CHEN Ramsey Theory and Classical Graph Ramsey Number
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Ramsey Theory and Classical Graph Ramsey Number

Mar 17, 2022

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Page 1: Ramsey Theory and Classical Graph Ramsey Number

Department of Mathematics, Nanjing University

Email: [email protected]

7 April, 2021

Yaojun CHEN

Ramsey Theory and Classical Graph Ramsey Number

Page 2: Ramsey Theory and Classical Graph Ramsey Number

OUTLINE

I. Originations of Ramsey Theory

II. Classical Graph Ramsey Number

Page 3: Ramsey Theory and Classical Graph Ramsey Number

What is Ramsey Theory ?

According to a 3500-year-old cuneiform text, an ancient Sumerian scholar once looked to the stars in the heavens and saw

a Lion

Page 4: Ramsey Theory and Classical Graph Ramsey Number

a BULL a SCORPION

Today, most stargazers would agree that the night sky appears to be filled with constellations in the shape of straight lines, rectangles and pentagons, and so on.Could it be that such geometric patterns arise from some unknown forces in the cosmos?

Page 5: Ramsey Theory and Classical Graph Ramsey Number

Mathematics provides a much more plausible explanation.

In 1928, an English mathematician, philosopher and economist F.P. Ramsey proved that such patterns are actually implicit in any large structure, no matter it is a group of stars or an array of pebbles!

F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc., 30(1930), 361-376.

Page 6: Ramsey Theory and Classical Graph Ramsey Number

1. Ramsey theorem

2. Schur theorem

3. Van der Waerden theorem

4. Erdős-Szekeres theorem

I. Originations of Ramsey Theory

Page 7: Ramsey Theory and Classical Graph Ramsey Number

1. Ramsey TheoremPARTY PUZZLE How many people does it take to form a group that always contains either n mutual acquaintances or n mutual strangers?

For it is known that at least 6 people is needed.

n=3,

How to prove it? One method is to list all conceivable combinations, check each one for an acquainted or unacquainted group of three.

Page 8: Ramsey Theory and Classical Graph Ramsey Number

Since we would have to check

combinations, this method is neither practical nor insightful.

Now, let points represent people in the party, a red edge joins people who are mutual acquainted, a blue edge joins people who are mutual strangers.

2(62) = 215 = 32768

1 2

3

45

6acquainted

strangers

Page 9: Ramsey Theory and Classical Graph Ramsey Number

1

23456

If {2,3,4} are mutual strangers, then the conclusion holds.

If some two people of {2,3,4} who are mutual acquainted, say 2 and 3 are mutual acquainted, then {1,2,3} are acquainted.

On the other hand, 5 people with relations illustrated in the left shows that 5 is not sufficient.

Page 10: Ramsey Theory and Classical Graph Ramsey Number

A complete graph : a graph on N vertices, any two vertices are connected by an edge.

KN

u v u vG G

Page 11: Ramsey Theory and Classical Graph Ramsey Number

General Form of PARTY PUZZLE: Given positive integers p and q, How large can guarantee that any 2-edge colorings of a complete graph with red and blue, either contains a red , or a blue ? F.P. Ramsey proved that such integer N exists! Let R(p,q) be the least integer N such that any 2-edge colorings of a complete graph with red and blue, either contains a red , or a blue .

p, q≥2.N=N(p, q)

KNKN Kp Kq

KNKN Kp Kq

It is easy to see that R(p,2)=p, R(2,q)=q and R(p,q)=R(q,p).

Page 12: Ramsey Theory and Classical Graph Ramsey Number

Theorem 1. For any two integers p,q≥2, R(p,q) exists and if p,q≥3, then the following holds:

R(p, q)≤R(p−1,q)+R(p, q−1) .

Let N=R(p−1,q)+R(p, q−1) .

0

21

R(p−1,q)

R(p−1,q)+1

N−1

KN

0

21

R(p,q−1)

R(p,q−1)+1

N−1

KN

Page 13: Ramsey Theory and Classical Graph Ramsey Number

General Form of Ramsey TheoremTheorem 2. For any two given positive integers and there is such that for any and any k-colorings of there exists some i with and a -set such that are in color i.

r, k,q1, q2, . . . , qr ≥r, N=N(q1, q2, . . . , qk),

n≥N, [n](r),1≤ i≤k qi Si ⊆ [n]

S(r)i

denotes all r-subsets of .[n]={1,2,...,n}; S(r)i Si

The smallest integer N satisfies Theorem 2 is called Ramsey number, write as . R(r)(n1, n2, . . . , nk)If then Theorem 2 is drawer principle. If then , that is, multiple colors Ramsey number.

r =1,r =2, R(2)(n1, n2, . . . , nk)=R(n1, n2, . . . , nk)

Page 14: Ramsey Theory and Classical Graph Ramsey Number

2. Schur TheoremThe following result, due to Schur (1916), which is viewed as one of the originations of Ramsey theory:

Theorem 3. For any given integer k, there exists an N, such that if then for any k-colorings of there must have of the same color such that

n≥N, [n],x, y, z∈ [n]

x + y = z .Let be the least possible value of N in Theorem 3.

is called Schur number. Some known Schur numbers up to now: The value of is determined by computer in 1965.

SkSk

S1 =2, S2 =5, S3 =14, S4 =45.S4

Page 15: Ramsey Theory and Classical Graph Ramsey Number

Schur theorem can be proved by Ramsey theorem.For any given integer k, take

Let be any partition of . For any 2-subset of , we color with if Thus, we get a k-colorings of all edges.

S1, S2, . . . , Sk [N]{i, j} [N] {i, j} ℓ

| i−j |∈Sℓ .By Ramsey theorem, has a 3-subset such that all of its 2-subsets receive the same color.

[N](2) {a, b, c}

Assume and Clearly, for some with , and

a>b>c, x=a−b, y=b−c, z=a−c .x, y, z∈Sℓ ℓ 1≤ℓ≤k

x + y = z .

N = R(2)k

(3,3,...,3) .

Page 16: Ramsey Theory and Classical Graph Ramsey Number

a−c∈S t

a−b∈St

1 2

N

j i

If color the edge with , | j−i |∈Sℓ, ij ℓ 1≤ℓ≤k .

a b

c

b−c∈

S t

j−i∈Sℓ

Let and then x=a−b, y=b−c z=a−c, x + y = z .

Page 17: Ramsey Theory and Classical Graph Ramsey Number

3. Van der Waerden TheoremVan der Waerden (1928) proved the following.Theorem 4. For any given positive integers there exists a positive integer such that for any k-colorings of , has an arithmetic progressions with terms of the same color.

ℓ, k,W=W(ℓ, k)

[W] [W]ℓ

The least possible value of in Theorem 4, is called Van der Waerden number. Some known Van der Waerden numbers:

W(ℓ, k)

W(3,2)=9, W(3,3)=27, W(3,4)=76,W(4,2)=35, W(5,2)=178.

Page 18: Ramsey Theory and Classical Graph Ramsey Number

Proof of W(3,2)=9.

1 2 3 4 5 6 7 8 9

Case 1. 4 and 6 has the same color, say in red.

4 652 8Case 2. 4 and 6 are in different colors.

1 2 3 4 5 6 7 8 94 652 83 71 9

1 2 3 4 5 6 7 8 94 652 83 71 9

Page 19: Ramsey Theory and Classical Graph Ramsey Number

4. Erdős-Szekeres TheoremErdős and Szekeres (1935) published the following:

Theorem 5. Let m be a positive integer. Then there exists a positive integer N, such that for any N points lie in a plane so that no three points form a straight line, there have m points form a convex m-polygon.

For m=4, N=5.Choose three points, say A,B,C such that ∠ABC<1800 and the other two points D, E lie in ∠ABC.

A

B

C

D

A

B

C

DE

Page 20: Ramsey Theory and Classical Graph Ramsey Number

Let N(m) be the smallest integer in Theorem 5. We have known that

No other values of N(m) are known up to now.

N(3) = 3,N(4) = 5,N(5) = 9.

Erdős conjectured that

The conjecture is true for m = 3,4,5.

N(m) = 1 + 2m−2 .

Page 21: Ramsey Theory and Classical Graph Ramsey Number

II. Classical Graph Ramsey Number

Page 22: Ramsey Theory and Classical Graph Ramsey Number

A challenging problem is to calculate R(5,5).

Up to know, all known values of classical Ramsey numbers R(p,q) are as follows:

It is very difficult to evaluate classical Ramsey numbers both exactly and asymptotically. Most of the values in the table are obtained with help of computers.

p 3 3 3 3 3 3 3 4 4q 3 4 5 6 7 8 9 4 5

R(p,q) 6 9 14 18 23 28 36 18 25

It is known that 43≤R(5,5)≤48 .

Page 23: Ramsey Theory and Classical Graph Ramsey Number

0 12

3

4

5

6

789

10

11

12

13

14

1516

Page 24: Ramsey Theory and Classical Graph Ramsey Number

Bounds for Classical Ramsey numbers

Erdős’ lower bound (1947): for R(p, p)>2

p2 p≥3.

The number of all red-blue edge colorings of isKn

The number of colorings with monochromatic is at most

Kp

2(n2) .

(np) ⋅ 2 ⋅ 2(n

2)−(p2) .

Page 25: Ramsey Theory and Classical Graph Ramsey Number

If

then there exists a red-blue colorings such that has no monochromatic .Kn Kp

(np) ⋅ 2(n

2)−(p2)+1 < 2(n

2),

By the definition of , we haveR(p, p)

R(p, p)>n .

Page 26: Ramsey Theory and Classical Graph Ramsey Number

It is not difficult to show that if thenn ≤ 2p2,

Thus,

By the argument above, we have

(np) <

np

2p−1≤ 2

p22 −p+1

= 2 12 p(p − 1)−1 ⋅ 2− p

2 +2 ≤ 2(p2)−1 .

(np) ⋅ 2(n

2)−(p2)+1 < 2(n

2) .

R(p, p) > 2p2 .

Page 27: Ramsey Theory and Classical Graph Ramsey Number

Probabilistic Method:

Color the edges of a complete graph randomly. That is, color each edge red with probability 1/2, and blue with probability 1/2. Since the probability that a given copy of has all edges red is

KN

Kp

2−(p2),

the expected number of red copies of isKp

2−(p2)(N

p ) .

Page 28: Ramsey Theory and Classical Graph Ramsey Number

Similarly, the expected number of blue copies of isKp

2−(p2)(N

p ) .

Therefore, the expected number of monochromatic copies of isKp

21−(p2)(N

p ) .

Since

21−(p2)(N

p ) ≤ 21−(p2) ( eN

p )p

,

Page 29: Ramsey Theory and Classical Graph Ramsey Number

take

21−(p2)(N

p ) ≤ 21−(p2) ( eN

p )p

< 1.

N=(1 − o(1)) p

2e ( 2)p,

we have

This implies

R(p, p)≥(1 − o(1)) p

2e ( 2)p

.

Page 30: Ramsey Theory and Classical Graph Ramsey Number

Spencer’s lower bound (1975):

R(p, p)≥(1 − o(1)) 2pe ( 2)

p.

Problem 1. Does there exist a positive constant such that

ε

R(p, p)≥ (1+ε)2pe ( 2)

p

for all sufficiently large ?p

Page 31: Ramsey Theory and Classical Graph Ramsey Number

Erdős and Szekeres’ upper bound (1935):

R(p + 1,q + 1) ≤ (p + qp ) .

If then it is easy to see the result holds. Since we have

p + q ≤ 3,R(p + 1,q + 1)≤R(p, q + 1)+R(p + 1,q),

R(p + 1,q + 1) ≤ (p + q − 1p − 1 )+(p + q − 1

p )= (p + q

p ) .

Page 32: Ramsey Theory and Classical Graph Ramsey Number

Graham and Rödl’s upper bound (1987):

R(p + 1,q + 1) ≤(p + q

p )log log (p + q)

.

It is believed that this upper bound is far more being satisfied!

is called diagonal Ramsey number if and off-diagonal Ramsey number otherwise. For , more advances are obtained recently.

R(p, q)p = q,

R(p, p)

Page 33: Ramsey Theory and Classical Graph Ramsey Number

Conlon’s upper bound (2009):

where c is a constant.

R(p + 1,p + 1) ≤ p−clog p/ log log p(2pp ) .

Sah’s upper bound (2020, arXiv:2005.09251):

where c is a constant.

R(p + 1,p + 1)≤e−c(log p)2(2pp ) .

Page 34: Ramsey Theory and Classical Graph Ramsey Number

It is known that:

R(p, p) ≤ (2p − 2p − 1 ) = O ( 4p

p ) .

Problem 2. Does

exist?

limp→∞

p R (p, p)

2 ≤ limp→∞

p R (p, p) ≤ 4.

If true, then

Page 35: Ramsey Theory and Classical Graph Ramsey Number

For off-diagonal Ramsey number , if is fixed and , then Erdős and Szekeres’ upper bound implies

R(p, q)p q → ∞

R(p, q)≤qp−1 .

Ajtai, Komlós and Szemerédi improved the bound

R(p, q)≤cpqp−1

(log q)p−2 .

where is a constant dependent on cp p .

Page 36: Ramsey Theory and Classical Graph Ramsey Number

When it says there is a constant such thatp = 3, c

R(3,q)≤cq2

log q.

Kim’s lower bound for (1995)::R(3,q)

R(3,q)= Θ(q2/ log q) .

R(3,q)≥cq2

log q,

where is a constant. Thus, the asymptotical order of is:

cR(3,q)

Page 37: Ramsey Theory and Classical Graph Ramsey Number

Thank you for your attention!