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Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Orthogonal polarity graphs and Sidon sets

Michael Tait

University of California-San Diego mtait@math.ucsd.edu Joint work with Craig Timmons

June 5, 2014

Michael Tait (UCSD) June 5, 2014 1 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Let (P, L, I) be a set of points, a set of lines, and a set of incidences in P × L.

Michael Tait (UCSD) June 5, 2014 2 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Let (P, L, I) be a set of points, a set of lines, and a set of incidences in P × L.

Definition

A polarity is a map π : P ∪ L → P ∪ L with the following properties:

Michael Tait (UCSD) June 5, 2014 2 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Let (P, L, I) be a set of points, a set of lines, and a set of incidences in P × L.

Definition

A polarity is a map π : P ∪ L → P ∪ L with the following properties:

1 π(P) = L and π(L) = P. Michael Tait (UCSD) June 5, 2014 2 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Let (P, L, I) be a set of points, a set of lines, and a set of incidences in P × L.

Definition

A polarity is a map π : P ∪ L → P ∪ L with the following properties:

1 π(P) = L and π(L) = P. 2 π2 = id. Michael Tait (UCSD) June 5, 2014 2 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Let (P, L, I) be a set of points, a set of lines, and a set of incidences in P × L.

Definition

A polarity is a map π : P ∪ L → P ∪ L with the following properties:

1 π(P) = L and π(L) = P. 2 π2 = id. 3 For p ∈ P and l ∈ L, (π(l), π(p)) ∈ I if and only if

(p, l) ∈ I.

Michael Tait (UCSD) June 5, 2014 2 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Given a geometry (P, L, I) and a polarity π, one can construct a polarity graph.

Michael Tait (UCSD) June 5, 2014 3 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Given a geometry (P, L, I) and a polarity π, one can construct a polarity graph. V (Gπ) = P

Michael Tait (UCSD) June 5, 2014 3 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Given a geometry (P, L, I) and a polarity π, one can construct a polarity graph. V (Gπ) = P E(Gπ) = {{p, q} : p = q ∈ P, (p, π(q)) ∈ I}.

Michael Tait (UCSD) June 5, 2014 3 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Polarities

Given a geometry (P, L, I) and a polarity π, one can construct a polarity graph. V (Gπ) = P E(Gπ) = {{p, q} : p = q ∈ P, (p, π(q)) ∈ I}. If (p, π(p)) ∈ I then p is called an absolute point.

Michael Tait (UCSD) June 5, 2014 3 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

Let P be the one-dimensional subspaces of F3

q and L be the

two-dimensional subspaces.

Michael Tait (UCSD) June 5, 2014 4 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

Let P be the one-dimensional subspaces of F3

q and L be the

two-dimensional subspaces. Define I by containment. i.e. (P, L, I) is a finite projective plane of order q.

Michael Tait (UCSD) June 5, 2014 4 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

Let P be the one-dimensional subspaces of F3

q and L be the

two-dimensional subspaces. Define I by containment. i.e. (P, L, I) is a finite projective plane of order q. Define a map π that sends points and lines to their

• rthogonal complements.

Michael Tait (UCSD) June 5, 2014 4 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

Let P be the one-dimensional subspaces of F3

q and L be the

two-dimensional subspaces. Define I by containment. i.e. (P, L, I) is a finite projective plane of order q. Define a map π that sends points and lines to their

• rthogonal complements. π is a polarity.

Michael Tait (UCSD) June 5, 2014 4 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

Let P be the one-dimensional subspaces of F3

q and L be the

two-dimensional subspaces. Define I by containment. i.e. (P, L, I) is a finite projective plane of order q. Define a map π that sends points and lines to their

• rthogonal complements. π is a polarity.

(x0, x1, x2) ∼ (y0, y1, y2) when x0y0 + x1y1 + x2y2 = 0.

Michael Tait (UCSD) June 5, 2014 4 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

This particular polarity graph was constructed by Erd˝

• s,

R´ enyi, and S´

• s (1966) and by Brown (1966) in relation to a

problem in extremal graph theory.

Michael Tait (UCSD) June 5, 2014 5 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

This particular polarity graph was constructed by Erd˝

• s,

R´ enyi, and S´

• s (1966) and by Brown (1966) in relation to a

problem in extremal graph theory. We will call this graph the Erd˝

• s-R´

enyi polarity graph and denote it by ERq.

Michael Tait (UCSD) June 5, 2014 5 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

This particular polarity graph was constructed by Erd˝

• s,

R´ enyi, and S´

• s (1966) and by Brown (1966) in relation to a

problem in extremal graph theory. We will call this graph the Erd˝

• s-R´

enyi polarity graph and denote it by ERq. ERq has q2 + q + 1 vertices and 1

2q(q + 1)2 edges.

Michael Tait (UCSD) June 5, 2014 5 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A polarity graph

This particular polarity graph was constructed by Erd˝

• s,

R´ enyi, and S´

• s (1966) and by Brown (1966) in relation to a

problem in extremal graph theory. We will call this graph the Erd˝

• s-R´

enyi polarity graph and denote it by ERq. ERq has q2 + q + 1 vertices and 1

2q(q + 1)2 edges.

ERq does not contain C4 as a subgraph.

Michael Tait (UCSD) June 5, 2014 5 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

ER4

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Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

ER8

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Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

ER16

Michael Tait (UCSD) June 5, 2014 8 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs

The study of C4 free graphs with many edges has a rich history in extremal combinatorics.

Michael Tait (UCSD) June 5, 2014 9 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs

The study of C4 free graphs with many edges has a rich history in extremal combinatorics.

Definition

The Tur´ an number for C4 is the maximum number of edges in an n-vertex graph that does not contain C4 as a subgraph. This quantity is denoted by ex(n, C4).

Michael Tait (UCSD) June 5, 2014 9 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs

The study of C4 free graphs with many edges has a rich history in extremal combinatorics.

Definition

The Tur´ an number for C4 is the maximum number of edges in an n-vertex graph that does not contain C4 as a subgraph. This quantity is denoted by ex(n, C4). ERq implies that ex(q2 + q + 1, C4) ≥ 1

2q(q + 1)2.

Michael Tait (UCSD) June 5, 2014 9 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs

A counting argument of K˝

• v´

ari, S´

• s, and Tur´

an (1954) gives that ex(n, C4) ≤ 1 2n3/2 + 1 2n.

Michael Tait (UCSD) June 5, 2014 10 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs

A counting argument of K˝

• v´

ari, S´

• s, and Tur´

an (1954) gives that ex(n, C4) ≤ 1 2n3/2 + 1 2n. Combined with the lower bound from ERq, ex(n, C4) ∼ 1

2n3/2.

Michael Tait (UCSD) June 5, 2014 10 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs: exact results

Computer search gives ex(n, C4) for n ≤ 31

Michael Tait (UCSD) June 5, 2014 11 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs: exact results

Computer search gives ex(n, C4) for n ≤ 31

Theorem (F¨ uredi)

Let q be a prime power. Then ex(q2 + q + 1, C4) = 1 2q(q + 1)2. Furthermore if q is even or if q > 13, then ERq is the unique extremal graph.

Michael Tait (UCSD) June 5, 2014 11 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

C4 free graphs: exact results

Computer search gives ex(n, C4) for n ≤ 31

Theorem (F¨ uredi)

Let q be a prime power. Then ex(q2 + q + 1, C4) = 1 2q(q + 1)2. Furthermore if q is even or if q > 13, then ERq is the unique extremal graph.

Theorem (Firke, Kosek, Nash, Williford 2013)

Let q be an even prime power. Then ex(q2 + q, C4) = 1 2q(q + 1)2 − q

Michael Tait (UCSD) June 5, 2014 11 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Sidon sets

Other values of n?

Michael Tait (UCSD) June 5, 2014 12 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Sidon sets

Other values of n? We use Sidon sets to study this problem.

Michael Tait (UCSD) June 5, 2014 12 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Sidon sets

Other values of n? We use Sidon sets to study this problem.

Definition

Given an abelian group Γ, a Sidon set A is a set A ⊂ Γ such that a, b, c, d ∈ A and a + b = c + d implies that {a, b} = {c, d}.

Michael Tait (UCSD) June 5, 2014 12 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Erd˝

• s conjecture

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Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Sidon Sets

Bose-Chowla Sidon Sets:Let q be a prime power and θ = F∗

q2.

A = {a ∈ Zq2−1 : θa − θ ∈ Fq}.

Michael Tait (UCSD) June 5, 2014 14 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ.

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ. V (Gq,θ) = Zq2−1

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ. V (Gq,θ) = Zq2−1 x ∼ y if x + y ∈ A.

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ. V (Gq,θ) = Zq2−1 x ∼ y if x + y ∈ A. i.e. Gq,θ is the Cayley sum graph of the Bose-Chowla Sidon set A.

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ. V (Gq,θ) = Zq2−1 x ∼ y if x + y ∈ A. i.e. Gq,θ is the Cayley sum graph of the Bose-Chowla Sidon set A. If x + x ∈ A, x is called an absolute point.

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ. V (Gq,θ) = Zq2−1 x ∼ y if x + y ∈ A. i.e. Gq,θ is the Cayley sum graph of the Bose-Chowla Sidon set A. If x + x ∈ A, x is called an absolute point. Gq,θ is almost q-regular.

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ

We use a Bose-Chowla Sidon set A to create a graph Gq,θ. V (Gq,θ) = Zq2−1 x ∼ y if x + y ∈ A. i.e. Gq,θ is the Cayley sum graph of the Bose-Chowla Sidon set A. If x + x ∈ A, x is called an absolute point. Gq,θ is almost q-regular. Absolute points have degree q − 1

Michael Tait (UCSD) June 5, 2014 15 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ is C4 free

Michael Tait (UCSD) June 5, 2014 16 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ is C4 free

Michael Tait (UCSD) June 5, 2014 16 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ is C4 free x + y + z + w = a + c = b + d

Michael Tait (UCSD) June 5, 2014 16 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Gq,θ is C4 free x + y + z + w = a + c = b + d = ⇒ {a, c} = {b, d}.

Michael Tait (UCSD) June 5, 2014 16 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

We can either add to or remove from Gq,θ to get lower bounds for ex(n, C4).

Michael Tait (UCSD) June 5, 2014 17 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

We can either add to or remove from Gq,θ to get lower bounds for ex(n, C4). This technique of strategically deleting vertices has been used to obtain lower bounds on ex(n, C4).

Michael Tait (UCSD) June 5, 2014 17 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

We can either add to or remove from Gq,θ to get lower bounds for ex(n, C4). This technique of strategically deleting vertices has been used to obtain lower bounds on ex(n, C4).

Theorem (Abreu, Balbuena, Labbate 2010)

Let q be a prime power. Then ex(q2 − q − 2, C4) ≥

• 1

2q3 − q2 − q 2 + 1

if q is odd

1 2q3 − q2

if q is even

Michael Tait (UCSD) June 5, 2014 17 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

We can either add to or remove from Gq,θ to get lower bounds for ex(n, C4). This technique of strategically deleting vertices has been used to obtain lower bounds on ex(n, C4).

Theorem (Abreu, Balbuena, Labbate 2010)

Let q be a prime power. Then ex(q2 − q − 2, C4) ≥

• 1

2q3 − q2 − q 2 + 1

if q is odd

1 2q3 − q2

if q is even They conjectured these bounds are tight.

Michael Tait (UCSD) June 5, 2014 17 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Their conjecture is false for q odd.

Michael Tait (UCSD) June 5, 2014 18 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Their conjecture is false for q odd.

Theorem (MT, Timmons)

Let q be an odd prime power. Then ex(q2 − q − 2, C4) ≥ 1 2q3 − q2 − O

• q3/4

.

Michael Tait (UCSD) June 5, 2014 18 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Their conjecture is false for q odd.

Theorem (MT, Timmons)

Let q be an odd prime power. Then ex(q2 − q − 2, C4) ≥ 1 2q3 − q2 − O

• q3/4

. Our best lower bound matched theirs when q is even.

Michael Tait (UCSD) June 5, 2014 18 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Proof Idea: Remove a “good” subset of vertices from Gq,θ.

Michael Tait (UCSD) June 5, 2014 19 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Proof Idea: Remove a “good” subset of vertices from Gq,θ. A subset is “good” if it contains many absolute points and/or many edges.

Michael Tait (UCSD) June 5, 2014 19 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

We are hopeful that we can use Gq,θ to solve other problems about ERq.

Michael Tait (UCSD) June 5, 2014 20 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

We are hopeful that we can use Gq,θ to solve other problems about ERq.

Theorem (MT, Timmons)

For q even or q ≥ 15 a prime power, Gq,θ is a large induced subgraph of ERq.

Michael Tait (UCSD) June 5, 2014 20 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Proof Idea: Using the definition of the Bose-Chowla Sidon set gives us information about how Gq,θ looks.

Michael Tait (UCSD) June 5, 2014 21 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Proof Idea: Using the definition of the Bose-Chowla Sidon set gives us information about how Gq,θ looks. Add q + 2 vertices and the “correct” number of edges, making sure no C4 is created

Michael Tait (UCSD) June 5, 2014 21 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

A lower bound

Proof Idea: Using the definition of the Bose-Chowla Sidon set gives us information about how Gq,θ looks. Add q + 2 vertices and the “correct” number of edges, making sure no C4 is created Now we have a graph on q2 + q + 1 vertices with 1

2q(q + 1)2

• edges. By F¨

uredi, it must be ERq.

Michael Tait (UCSD) June 5, 2014 21 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

Corollary

Let q be large enough. Then the Petersen graph is a subgraph of ERq.

Michael Tait (UCSD) June 5, 2014 22 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

Corollary

Let q be large enough. Then the Petersen graph is a subgraph of ERq.

Corollary

Let q be large enough. Then there is a subgraph of ERq of constant size with chromatic number at least 4.

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Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

What subgraphs does ERq contain?

Michael Tait (UCSD) June 5, 2014 23 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

What subgraphs does ERq contain? What is the largest triangle-free induced subgraph of ERq?

Michael Tait (UCSD) June 5, 2014 23 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Using Gq,θ

What subgraphs does ERq contain? What is the largest triangle-free induced subgraph of ERq? What is the independence number of ERq?

Michael Tait (UCSD) June 5, 2014 23 / 24

Michael Tait Polarity graphs Sidon sets and C4 free graphs Results Open Problems

Thank You

Michael Tait (UCSD) June 5, 2014 24 / 24