# Extending Fisher’s inequality to gfarr/research/slides/Horsley-May2016Fisher... ·...

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Extending Fishers inequality to coverings

Daniel Horsley (Monash University, Australia)

Introduction 1Designs and Fishers inequality

(v, k, )-designs

A collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

(v, k, )-designsA collection of k-subsets (blocks) of a v-set (of points) such that every pairof points appears together in exactly blocks.

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

12

3

4

56

7

8

9

A (9,3,1)-design with 12 blocks

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

12

3

4

56

7

8

9

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Eventually all exist

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Eventually all exist

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Eventually all exist

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.

Equivalently,I with b < v; orI with r < k.

Symmetric designs have v = k(k1) + 1 (or b = v or r = k).

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.

Equivalently,I with b < v; orI with r < k.

Symmetric designs have v = k(k1) + 1 (or b = v or r = k).

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.

Equivalently,I with b < v; orI with r < k.

Symmetric designs have v = k(k1) + 1 (or b = v or r = k).

Necessary conditions for a design to exist

Obvious necessary conditions: If there exists a (v, k, )-design then

(1) r = (v 1)k 1 is an integer;

(2) b = rvk is an integer.

Fishers inequality (1940): There is no (v, k, )-design with v < k(k1) + 1.

Equivalently,I with b < v; orI with r < k.

Symmetric designs have v = k(k1) + 1 (or b = v or r = k).

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Possible (v,10,1)-designs

46 55 91 100 136 145 181 190v0

100

200

300

400

b

Incidence matrix arithmetic

Consider the incidence matrix of our (9,3,1)-design.

12 blocks

9po

ints

1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1

Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.

12 blocks

9po

ints

1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1

Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.

12 blocks

9po

ints

1 0 0 1 0 0 1 0 0 1 0 01 1 0 0 1 0 0 1 0 0 0 01 0 1 0 0 1 0 0 0 0 1 00 1 0 1 0 0 0 0 1 0 1 00 0 1 0 1 0 1 0 1 0 0 00 0 0 0 0 1 0 1 1 1 0 00 0 1 1 0 0 0 1 0 0 0 10 0 0 0 1 0 0 0 0 1 1 10 1 0 0 0 1 1 0 0 0 0 1

12

3

4

56

7

8

9

Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.

12 blocks

9po

ints

Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.

12 blocks

9po

ints

= A

Incidence matrix arithmeticConsider the incidence matrix of our (9,3,1)-design.

12 blocks

9po

ints

= A

AAT =

4 1 1 1 1 1 1 1 11 4 1 1 1 1 1 1 11 1 4 1 1 1 1 1 11 1 1 4 1 1 1 1 11 1 1 1 4 1 1 1 11 1 1 1 1 4 1 1 11 1 1