Flag Algebras (hopefully simple basics) Bernard Lidick y - - PowerPoint PPT Presentation

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag Algebras (hopefully simple basics)

Bernard Lidick´ y

University of Illinois at Urbana-Champaign

Apr 29, 2014

Flag algebras First try for Mantel More automatic approach Additional constraints

Outline

• Flag Algebras definitions
• First try for Mantel’s theorem
• More automatic approach
• Additional constraints

2

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag Algebras definitions

• Basic definitions
• Some identities

3

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras

Seminal paper:

• A. Razborov, Flag Algebras, Journal of Symbolic Logic 72 (2007),

1239–1282. David P. Robbins Prize by AMS for Razborov in 2013

4

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras

Seminal paper:

• A. Razborov, Flag Algebras, Journal of Symbolic Logic 72 (2007),

1239–1282. David P. Robbins Prize by AMS for Razborov in 2013 Applications to oriented graphs, hypergraphs, crossing number of complete bipartite graphs, geometry,. . .

Theorem (Hatami,Hladk´

y,Kr´ a , l,Norine,Razborov 2011; Grzesik 2011)

The number of C5’s in a triangle-free graph on n vertices is at most (n/5)5.

n 5 n 5 n 5 n 5 n 5 4

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra definitions

Let G be a 2-edge-colored complete graph on n vertices. The probability that three random vertices in G span a red triangle.

5

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra definitions

Let G be a 2-edge-colored complete graph on n vertices. The probability that three random vertices in G span a red triangle. The probability that three random vertices in G span a triangle with one red and two blue edges.

5

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra definitions

Let G be a 2-edge-colored complete graph on n vertices. The probability that three random vertices in G span a red triangle. The probability that three random vertices in G span a triangle with one red and two blue edges.

v

The probability that a random vertex other than v is connected to v ∈ V (G) by a red edge, i.e., the red degree of v divided by n − 1.

5

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra definitions

Let G be a 2-edge-colored complete graph on n vertices. The probability that three random vertices in G span a red triangle. The probability that three random vertices in G span a triangle with one red and two blue edges.

v

The probability that a random vertex other than v is connected to v ∈ V (G) by a red edge, i.e., the red degree of v divided by n − 1. + =

5

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra definitions

Let G be a 2-edge-colored complete graph on n vertices. The probability that three random vertices in G span a red triangle. The probability that three random vertices in G span a triangle with one red and two blue edges.

v

The probability that a random vertex other than v is connected to v ∈ V (G) by a red edge, i.e., the red degree of v divided by n − 1. + = 1

5

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra definitions

Let G be a 2-edge-colored complete graph on n vertices. The probability that three random vertices in G span a red triangle. The probability that three random vertices in G span a triangle with one red and two blue edges.

v

The probability that a random vertex other than v is connected to v ∈ V (G) by a red edge, i.e., the red degree of v divided by n − 1. + = 1 Type - flag induced by labeled vertices

1 2

Flag

5

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

Let G be a 2-edge-colored complete graph on n vertices. + + + = 1

6

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

Let G be a 2-edge-colored complete graph on n vertices. = 3 3 + 2 3 + 1 3 + 0 3

7

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

Let G be a 2-edge-colored complete graph on n vertices. = + 2 3 + 1 3

7

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

v × v = v ?

+ o(1) =

v + v + o(1)

• (1) as |V (G)| → ∞
• (1) will be omitted on next slides

8

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

v × v = v ?

+ o(1) =

v + v + o(1) v × v = 1

2

v ?

+ o(1) = 1 2

v + 1

2

v + o(1)

• (1) as |V (G)| → ∞
• (1) will be omitted on next slides

8

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

v × v = v ?

+ o(1) =

v + v + o(1) v × v = 1

2

v ?

+ o(1) = 1 2

v + 1

2

v + o(1)

• (1) as |V (G)| → ∞
• (1) will be omitted on next slides

Unordered Ordered

8

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

1 3 = 1 |V (G)|

• v∈V (G)

v

9

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

1 3 = 1 |V (G)|

• v∈V (G)

v

n 3

• =
• v∈V (G)

v

n − 1 2

• 9

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebra identities

1 3 = 1 |V (G)|

• v∈V (G)

v

= 1 |V (G)|

• v∈V (G)

v

n 3

• =
• v∈V (G)

v

n − 1 2

• 9

Flag algebras First try for Mantel More automatic approach Additional constraints

First try for Mantel’s theorem

• How to use the equation to prove something
• Gives bounds as well as helps with extremal examples

10

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue .

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤

• 1 − 2 v

2

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤

• 1 − 2 v

2 =

• 1 − 4 v

+ 4

v + 4 v

• v ×

v = v + v

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤ 1 n

• v
• 1 − 2 v

2 = 1 n

• v
• 1 − 4 v

+ 4

v + 4 v

• 11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤ 1 n

• v
• 1 − 2 v

2 = 1 n

• v
• 1 − 4 v

+ 4

v + 4 v

• = 1 − 4

+ 4 3 + 4

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤ 1 n

• v
• 1 − 2 v

2 = 1 n

• v
• 1 − 4 v

+ 4

v + 4 v

• = 1 − 4

+ 4 3 = 2

3

+ 1

3

+

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤ 1 n

• v
• 1 − 2 v

2 = 1 n

• v
• 1 − 4 v

+ 4

v + 4 v

• = 1 − 4

+ 4 3 = 2

3

+ 1

3

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤ 1 n

• v
• 1 − 2 v

2 = 1 n

• v
• 1 − 4 v

+ 4

v + 4 v

• = 1 − 4

+ 4 3 = 1 − 2 − 2 3 2 = 4

3

+ 2

3

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 1

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue . Assume = 0. (We want to conclude ≤ 1

2.)

0 ≤ 1 n

• v
• 1 − 2 v

2 = 1 n

• v
• 1 − 4 v

+ 4

v + 4 v

• = 1 − 4

+ 4 3 = 1 − 2 − 2 3 ≤ 1 − 2 2 = 4

3

+ 2

3

11

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1)

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1)

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - stability for Mantel

Assume = 0 and = 1

• 2. Goal is

. 0 ≤ 1 − 2 − 2 3 + o(1) 0 ≤ −2 3 + o(1) Only and appear.

12

Flag algebras First try for Mantel More automatic approach Additional constraints

More automatic approach

• How to use computer to guess the right equation for you.

13

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue.

14

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue. Assume = 0. (We want to conclude ≤ 1

2.)

14

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue. Assume = 0. (We want to conclude ≤ 1

2.)

1 = + + +

14

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue. Assume = 0. (We want to conclude ≤ 1

2.)

1 = + +

14

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue. Assume = 0. (We want to conclude ≤ 1

2.)

1 = + + = 0 + 1 3 + 2 3

14

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue. Assume = 0. (We want to conclude ≤ 1

2.)

1 = + + = 0 + 1 3 + 2 3 ≤ 2 3

• +

+

• 14

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - example

Theorem (Mantel 1907)

A triangle-free graph contains at most 1

4n2 edges.

Assume edges are red and non-edges are blue. Assume = 0. (We want to conclude ≤ 1

2.)

1 = + + = 0 + 1 3 + 2 3 ≤ 2 3

• +

+

• ≤ 2

3

14

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 2

Assume = 0. (We want to conclude ≤ 1

2.)

= 0 + 1 3 + 2 3

15

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 2

Assume = 0. (We want to conclude ≤ 1

2.)

= 0 + 1 3 + 2 3 Idea: find c1, c2, c3 ∈ R such that for every graph G 0 ≤ c1 + c2 + c3 .

15

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 2

Assume = 0. (We want to conclude ≤ 1

2.)

= 0 + 1 3 + 2 3 Idea: find c1, c2, c3 ∈ R such that for every graph G 0 ≤ c1 + c2 + c3 . After summing together ≤ c1 + 1 3 + c2

• +

2 3 + c3

• and

≤ max

• (0 + c1) , 1

3 + c2, 2 3 + c3

• .

15

Flag algebras First try for Mantel More automatic approach Additional constraints

Example - Mantel’s theorem version 2

Assume = 0. (We want to conclude ≤ 1

2.)

= 0 + 1 3 + 2 3 Idea: find c1, c2, c3 ∈ R such that for every graph G 0 ≤ c1 + c2 + c3 . After summing together ≤ c1 + 1 3 + c2

• +

2 3 + c3

• and

≤ max

• (0 + c1) , 1

3 + c2, 2 3 + c3

• .

c3 < 0

15

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤

• v

,

v

a c c b

v

,

v

T a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤

• v

,

v

a c c b

v

,

v

T = a

v ?

+ b

v ?

+ 1 2c

v ?

+ 1 2c

v ?

a c c b

• 0 (matrix is positive semidefinite)

v × v = v ?

+ o(1)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤

• v

,

v

a c c b

v

,

v

T = a

v ?

+ b

v ?

+ 1 2c

v ?

+ 1 2c

v ?

a c c b

• 0 (matrix is positive semidefinite)

v × v = v ?

+ o(1)

v × v = 1

2

v ?

+ o(1) Unordered Ordered

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤

• v

,

v

a c c b

v

,

v

T = a

v ?

+ b

v ?

+ c

v ?

a c c b

• 0 (matrix is positive semidefinite)

v × v = v ?

+ o(1)

v × v = 1

2

v ?

+ o(1) Unordered Ordered

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤

• v

,

v

a c c b

v

,

v

T = a

v ?

+ b

v ?

+ c

v ?

a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤ 1 n

• v
• v

,

v

a c c b

v

,

v

T = 1 n

• v

a

v ?

+ b

v ?

+ c

v ?

a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤ 1 n

• v
• v

,

v

a c c b

v

,

v

T = 1 n

• v

a

v ?

+ b

v ?

+ c

v ?

= a + a + 2c 3 + b + 2c 3 + b a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤ 1 n

• v
• v

,

v

a c c b

v

,

v

T = 1 n

• v

a

v ?

+ b

v ?

+ c

v ?

= a + a + 2c 3 + b + 2c 3 a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - candidates for c1, c2, c3

0 ≤ 1 n

• v
• v

,

v

a c c b

v

,

v

T = 1 n

• v

a

v ?

+ b

v ?

+ c

v ?

= a + a + 2c 3 + b + 2c 3 c1 = a, c2 = a + 2c 3 , c3 = b + 2c 3 a c c b

• 0 (matrix is positive semidefinite)

16

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - using c1, c2, c3

= + 1 3 + 2 3 0 ≤ a + a + 2c 3 + b + 2c 3 a c c b

• 0 (matrix is positive semidefinite)

17

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - using c1, c2, c3

= + 1 3 + 2 3 0 ≤ a + a + 2c 3 + b + 2c 3 a c c b

• 0 (matrix is positive semidefinite)

17

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - using c1, c2, c3

= + 1 3 + 2 3 0 ≤ a + a + 2c 3 + b + 2c 3 ≤ max

• a, 1 + a + 2c

3 , 2 + b + 2c 3

• .

a c c b

• 0 (matrix is positive semidefinite)

17

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - using c1, c2, c3

= + 1 3 + 2 3 0 ≤ a + a + 2c 3 + b + 2c 3 ≤ max

• a, 1 + a + 2c

3 , 2 + b + 2c 3

• .

Try a c c b

• =
• 1/2

−1/2 −1/2 1/2

• .

17

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - using c1, c2, c3

= + 1 3 + 2 3 0 ≤ a + a + 2c 3 + b + 2c 3 ≤ max

• a, 1 + a + 2c

3 , 2 + b + 2c 3

• .

Try a c c b

• =
• 1/2

−1/2 −1/2 1/2

• .

It gives ≤ max 1 2, 1 6, 1 2

• = 1

2.

17

Flag algebras First try for Mantel More automatic approach Additional constraints

Flag algebras - optimizing a, b, c

≤ max

• a, 1 + a + 2c

3 , 2 + b + 2c 3

• (SDP)

                     Minimize d subject to a ≤ d

1+a+2c 3

≤ d

2+b+2c 3

≤ d

• a

c c b

• (SDP) can be solved on computers using CSDP or SDPA.

18

Flag algebras First try for Mantel More automatic approach Additional constraints

How to get stability?

In first try we got 0 ≤ 1 − 2 − 2

3

which gives stability.

19

Flag algebras First try for Mantel More automatic approach Additional constraints

How to get stability?

We got ≤ max 1 2, 1 6, 1 2

• = 1

2. which is ≤ 1 2 + 1 6 + 1 2

19

Flag algebras First try for Mantel More automatic approach Additional constraints

How to get stability?

≤ 1 2 + 1 6 + 1 2

19

Flag algebras First try for Mantel More automatic approach Additional constraints

How to get stability?

≤ 1 2 + 1 6 + 1 2 Suppose G is an extremal graph ( ≥ 1

2). Then

1 2 = ≤ 1 2 + 1 6 + 1 2 1 ≤ + 1 3 +

19

Flag algebras First try for Mantel More automatic approach Additional constraints

How to get stability?

≤ 1 2 + 1 6 + 1 2 Suppose G is an extremal graph ( ≥ 1

2). Then

1 2 = ≤ 1 2 + 1 6 + 1 2 1 ≤ + 1 3 + Combined with 1 = + + gives 0 ≤ −2 3

19

Flag algebras First try for Mantel More automatic approach Additional constraints

≤ max 1 2, 1 6, 1 2

• = 1

2 Tells us that that if

• = 1

2

• , then
• graphs with coefficients < 1

2 do not appear in the extremal

example

• subgraphs of extremal example should have 1

2

• gives possible subgraphs for extremal examples (if not known)
• having 1

2 does not mean it appears in extremal example

20

Flag algebras First try for Mantel More automatic approach Additional constraints

Additional constraints

• Adding more constraints
• Considering bigger (but still small) graphs may improve

bounds

21

Flag algebras First try for Mantel More automatic approach Additional constraints

Small experiment

Mantel        Maximize subject to = 0 Solution is 1

2.

22

Flag algebras First try for Mantel More automatic approach Additional constraints

Small experiment

Mantel        Maximize subject to = 0 Solution is 1

• 2. But what if

= p > 1

2?

22

Flag algebras First try for Mantel More automatic approach Additional constraints

Small experiment

Mantel        Maximize subject to = 0 Solution is 1

• 2. But what if

= p > 1

2?

       Minimize subject to ≥ p

22

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ p.

23

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ p.

Theorem (Razborov ’08)

≥ (t − 1)

• t − 2
• t(t − p(t + 1))

t +

• t(t − p(t + 1))

2 t2(t + 1)2 where t = ⌊1/(1 − p)⌋. Tight bound.

23

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ p.

Theorem (Razborov ’08)

≥ (t − 1)

• t − 2
• t(t − p(t + 1))

t +

• t(t − p(t + 1))

2 t2(t + 1)2 where t = ⌊1/(1 − p)⌋. Tight bound. Nontrivial application of FA. We will try simple approach for p = 0.6

23

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ p.

Theorem (Razborov ’08)

≥ (t − 1)

• t − 2
• t(t − p(t + 1))

t +

• t(t − p(t + 1))

2 t2(t + 1)2 where t = ⌊1/(1 − p)⌋. Tight bound. Nontrivial application of FA. We will try simple approach for p = 0.6 (but we will get worse bound). ≥ 0.1415009 . . . for p = 0.6 by Razborov

23

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6.

24

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6. = 0 + 0 + 0 + ≥ min{0, 0, 0, 1}

24

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6. = 0 + 0 + 0 + ≥ min{0, 0, 0, 1} = 0 + 1 3 + 2 3 +

24

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6. = 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0.6 ≤ = 0 + 1 3 + 2 3 +

24

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6. = 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0.6 ≤ = 0 + 1 3 + 2 3 + 1 = + + +

24

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6. = 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0.6 ≤ = 0 + 1 3 + 2 3 + 0.6 = 0.6 + 0.6 + 0.6 + 0.6

24

Flag algebras First try for Mantel More automatic approach Additional constraints

Minimize subject to ≥ 0.6. = 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0.6 ≤ = 0 + 1 3 + 2 3 + 0.6 = 0.6 + 0.6 + 0.6 + 0.6 0 ≤ −0.6 + 1 3 − 0.6

• +

2 3 − 0.6

• + 0.4

24

Flag algebras First try for Mantel More automatic approach Additional constraints

= 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0 ≤ −0.6 + 1 3 − 0.6

• +

2 3 − 0.6

• + 0.4

25

Flag algebras First try for Mantel More automatic approach Additional constraints

= 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0 ≤ −0.6 + 1 3 − 0.6

• +

2 3 − 0.6

• + 0.4

0 ≤ 1 n

• v
• v

,

v

a c c b

v

,

v

T 0 ≤ a + a + 2c 3 + b + 2c 3 +b

25

Flag algebras First try for Mantel More automatic approach Additional constraints

= 0 + 0 + 0 + ≥ min{0, 0, 0, 1} 0 ≥ −a − a + 2c 3 − b + 2c 3 −b 0 ≥ d

• 0.6

+

• 0.6 − 1

3

• +
• 0.6 − 2

3

• − 0.4
• 

 a c c b d   0 (matrix is positive semidefinite)

26

Flag algebras First try for Mantel More automatic approach Additional constraints

= 0 + 0 + 0 + 0 ≥ −a − a + 2c 3 − b + 2c 3 −b 0 ≥ d

• 0.6

+

• 0.6 − 1

3

• +
• 0.6 − 2

3

• − 0.4
• ≥ min
• 0.6d − a,
• 0.6 − 1

3

• d − a + 2c

3 ,

• 0.6 − 2

3

• d − b + 2c

3 , 1 − 0.4d − b

• 27

Flag algebras First try for Mantel More automatic approach Additional constraints

≥ min

• 0.6d − a,
• 0.6 − 1

3

• d − a + 2c

3 ,

• 0.6 − 2

3

• d − b + 2c

3 , 1 − 0.4d − b

• Solution from CSDP:

a = 6 × 0.1200006508849779385 a = 0.72 b = 6 × 0.05333290843810910981 b = 0.32 c = 6 × −0.07999989818128358521 c = −0.48 d = 1.400006454027185265 d = 1.4 ≥ min

• 0.12, 0.453, 0.12, 0.12
• = 0.12

28

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12?

Sample bigger graphs. Instead of 1 = + + + use 1 = + + + · · · +

29

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12?

Sample bigger graphs. Instead of 1 = + + + use 1 = + + + · · · + and include also M, P 0 0 ≤

• 1

2 , 1 2 , 1 2 , 1 2 T

M

• 1

2 , 1 2 , 1 2 , 1 2

0 ≤

• 1

2 , 1 2 , 1 2 , 1 2 T

P

• 1

2 , 1 2 , 1 2 , 1 2

29

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12?

Sample bigger graphs. Instead of 1 = + + + use 1 = + + + · · · + and include also M, P 0 0 ≤

• 1

2 , 1 2 , 1 2 , 1 2 T

M

• 1

2 , 1 2 , 1 2 , 1 2

0 ≤

• 1

2 , 1 2 , 1 2 , 1 2 T

P

• 1

2 , 1 2 , 1 2 , 1 2

This gives ≥ 0.127815 . . .

29

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12781. . . ?

Sample even bigger graphs. Use K5 instead of K4 Include even more types and flags.

30

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12781. . . ?

Sample even bigger graphs. Use K5 instead of K4 Include even more types and flags. This gives ≥ 0.1333333 = 2/15.

30

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12781. . . ?

Sample even bigger graphs. Use K5 instead of K4 Include even more types and flags. This gives ≥ 0.1333333 = 2/15. But using K6 or K7 does not help... still 2/15 K9 probably not computable.

30

Flag algebras First try for Mantel More automatic approach Additional constraints

How to improve 0.12781. . . ?

Sample even bigger graphs. Use K5 instead of K4 Include even more types and flags. This gives ≥ 0.1333333 = 2/15. But using K6 or K7 does not help... still 2/15 K9 probably not computable. Note: our method works for p ∈ {2

3, 3 4, 4 5, . . .}, i.e. cases

⌊1/(1 − p)⌋ = 1/(1 − p). Works for Goodman’s bound: ≥ 2p2 − p a = 2p2 b = 2p2 − 4p + 2 c = p(2p − 2) d = 4p − 1

30

Flag algebras First try for Mantel More automatic approach Additional constraints

• we described only plain method
• there are more ways to get equations

0 ≤ linear combination of flags

• actual flag algebra is defined using positive homomorphisms

31

Flag algebras First try for Mantel More automatic approach Additional constraints

Thank you for your attention!

If case you liked the talk, it is a copy of a talk of Florian Pfender.

• we described only plain method
• there are more ways to get equations

0 ≤ linear combination of flags

• actual flag algebra is defined using positive homomorphisms

31