Infinite designs -- a quick tourcamconf/talks/webb.pdfInﬁnite designs – a quick tour Bridget S...

Infinite designs – a quick tour

Bridget S WebbThe Open University

COMBINATORICS, ALGEBRA, AND MORE:A CONFERENCE IN CELEBRATION OF PETER CAMERON

QMUL, 10 July 2013

Bridget S Webb (The Open University) 1 / 29

Route Map

Definitions

A (FINITE) t-(v , k , λ) design is a

v-set of points V

with a collection B of k-subsets called blocks

such that

every t-subset of points is contained in precisely λ blocks

A (FINITE) t-(v , k , λ) design is a

v-set of points V

such that

every t-subset of points is contained in precisely λ blocks

The fano plane {0 1 2 3 4 5 6}

{0 1 3}, {1 2 4}, {2 3 5}{3 4 6}, {0 4 5}, {1 5 6}{0 2 6}

2-(7, 3, 1) design

{0 1 3}, {1 2 4}, {2 3 5}{3 4 6}, {0 4 5}, {1 5 6}{0 2 6}

2-(7, 3, 1) design

The affine plane{1 2 3 4 5 6 7 8 9}

{1 2 3}, {4 5 6}, {7 8 9}{1 4 7}, {2 5 8}, {3 6 9}{1 5 9}, {2 6 7}, {3 4 8}{1 6 8}, {2 4 9}, {3 5 7}

2-(9, 3, 1) design

{0 1 3}, {1 2 4}, {2 3 5}{3 4 6}, {0 4 5}, {1 5 6}{0 2 6}

2-(7, 3, 1) design

{1 2 3}, {4 5 6}, {7 8 9}{1 4 7}, {2 5 8}, {3 6 9}{1 5 9}, {2 6 7}, {3 4 8}{1 6 8}, {2 4 9}, {3 5 7}

2-(9, 3, 1) design

A 2-(v , 3, 1) design is a Steiner triple system

{0 1 3}, {1 2 4}, {2 3 5}{3 4 6}, {0 4 5}, {1 5 6}{0 2 6}

2-(7, 3, 1) design

{1 2 3}, {4 5 6}, {7 8 9}{1 4 7}, {2 5 8}, {3 6 9}{1 5 9}, {2 6 7}, {3 4 8}{1 6 8}, {2 4 9}, {3 5 7}

2-(9, 3, 1) design

A 2-(v , 3, 1) design is a Steiner triple system

More generally, a t-(v , k , 1) design is a Steiner system

Euclidean plane

This is a 2-(2ℵ0, 2ℵ0, 1) design

Euclidean plane

This is a 2-(2ℵ0, 2ℵ0, 1) design

Euclidean disc

Another 2-(2ℵ0, 2ℵ0, 1) design

A t-(v , k , Λ) design is a

a v-set V of points

|V\B| = k , for all B ∈ B, where k + k = v

For 0 ≤ i + j ≤ t , the cardinality λi ,j of the set of blocks containingall of i points x1, . . . xi and none of j points y1, . . . yj , depends onlyon i and j

no block contains another block

a v-set V of points

such that

a v-set V of points

such that

a v-set V of points

such that

a v-set V of points

such that

Λ = (λi ,j) is a (t + 1) × (t + 1) matrix

λt,0 = λ, λ1,0 = r and λ0,0 = b

a v-set V of points

such that

Λ = (λi ,j) is a (t + 1) × (t + 1) matrix

λt,0 = λ, λ1,0 = r and λ0,0 = b

0 < t ≤ k ≤ v ensures non-degeneracy(CAMERON, BSW 2002)

Route map

Wild (or less tame) examples...

Rado’s graph is a 2-(ℵ0, ℵ0, Λ) design

points: the vertices of the random graph

blocks: the maximal cliques and maximal cocliques

λi ,j = 2ℵ0 for all i + j ≤ 2

so, k = v and b = λ > v(CAMERON, BSW 2002)

Rado’s graph is a 2-(ℵ0, ℵ0, Λ) design

points: the vertices of the random graph

blocks: the maximal cliques and maximal cocliques

λi ,j = 2ℵ0 for all i + j ≤ 2

so, k = v and b = λ > v(CAMERON, BSW 2002)

There is a 2-(ℵ0, ℵ0, Λ) design with λ = 2 with two different resolutionsinto parallel classes (λi ,j = ℵ0 for all i < t)

ℵ0 parallel classes of size ℵ0

one parallel class of 4 blocks and the remaining ℵ0 ones of size ℵ0

(DANZIGER, HORSLEY, BSW 201?)

A design with more points than blocks : v > b

Points: unit circle

Blocks: For each rational rootof unity S

B1s blue blockB2s purple block

This is a 2-(2ℵ0, 2ℵ0, Λ) designwith λi ,j = ℵ0

So, b = r = λ = ℵ0

(CAMERON, BSW 2002)This is resolvable with

ℵ0 parallel classes of 2 blocks each

Tame (or less wild) examples...

When v is INFINITE, and t and λ are both FINITE:

λt,0 = λ

λi ,j = v , for all i < t , 0 ≤ i + j ≤ t

We can write t-(v , k , λ), as in the FINITE case, without ambiguity

These designs are generally well behaved:

Fisher’s Inequality b ≥ v holds since v = b

when k < v , these designs are necessarily resolvable(DANZIGER, HORSLEY, BSW 201?)

When v is INFINITE, and t and λ are both FINITE:

λt,0 = λ

λi ,j = v , for all i < t , 0 ≤ i + j ≤ t

We can write t-(v , k , λ), as in the FINITE case, without ambiguity

These designs are generally well behaved:

Fisher’s Inequality b ≥ v holds since v = b

when k < v , these designs are necessarily resolvable(DANZIGER, HORSLEY, BSW 201?)

Triangular lattice

Points: the lattice points

Blocks: equilateral triangles

Triangular lattice

A 2-(ℵ0, 3, 2) design

Countably infinite Steiner Triple system

−Points: Q ∪ {+∞,−∞}

Blocks:◮ (x , y , z) where x + y + z = 0

and x , y , z unequal◮ (−2x , x ,±∞) and

(−x , 2x ,±∞)◮ (0, +∞,−∞)

− 0Points: Q ∪ {+∞,−∞}

(−x , 2x ,±∞)◮ (0, +∞,−∞)

− 0Points: Q ∪ {+∞,−∞}

(−x , 2x ,±∞)◮ (0, +∞,−∞)

− 0Points: Q ∪ {+∞,−∞}

(−x , 2x ,±∞)◮ (0, +∞,−∞)

− 0Points: Q ∪ {+∞,−∞}

(−x , 2x ,±∞)◮ (0, +∞,−∞)

A 2-(ℵ0, 3, 1) design

(GRANNELL, GRIGGS, PHELAN 1987)

In contrast to the FINITE case, the existence problem forINFINITE t-designs is incomparably simpler — basically, they exist!

Existence with

k FINITE

Cyclic t-(ℵ0, k , λ) (KÖHLER 1977)

Large sets t-(∞, t + 1, 1) (GRANNELL, GRIGGS, PHELAN 1991)

Large sets t-(∞, k , 1) (CAMERON 1995)

t-fold transitive t-(ℵ0, t + 1, 1) (CAMERON 1984)

Uncountable family of rigid 2-(ℵ0, 3, 1) (FRANEK 1994)

k not necessarily FINITE

Any t-(∞, k , 1) can be extended (BEUTELSPACHER, CAMERON 1994)

Existence with t ≥ 2

k FINITE

Existence with t ≥ 2

k FINITE

Route map

Constructions

1 2 3, 1 4 5,

1 2 3, 1 4 5, 2 4 6, 3 4 7,

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12,

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12, 1 7 13, 2 7 14, 5 7 15 . . .

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12, 1 7 13, 2 7 14, 5 7 15 . . .

The result is a 2-(ℵ0, 3, 1) design

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12, 1 7 13, 2 7 14, 5 7 15 . . .

r -sparse (for all r ≥ 4) i.e. no (r , r + 2)-configurations

no finite subsystems

proper subsystems isomorphic to itself

subsystems intrinsically more complicated than itself

The pasch , a (4, 6)-configuration

(CHICOT, GRANNELL, GRIGGS, BSW 2009)

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12, 1 7 13, 2 7 14, 5 7 15 . . .

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12, 1 7 13, 2 7 14, 5 7 15 . . .

1 2 3, 1 4 5, 2 4 6, 3 4 7, 2 5 8, 3 5 9,

1 6 10, 3 6 11, 5 6 12, 1 7 13, 2 7 14, 5 7 15 . . .

Constructions

Cycle graph G(a,b) of a Steiner triple system, where abc is a block:

vertices V\{a, b, c}

a-coloured edge xy if axy is a block

b-coloured edge xy if bxy is a block

A uniform Steiner triple system has all cycle graphs isomorphic

A perfect Steiner triple system has each cycle graph a single cycle(or two-way infinite path)

Only finitely many FINITE perfect systems are known

Free construction of a perfect Steiner triple system

Start with a partial Steiner triple system◮ no cycle graph G(a, b) containing a finite cycle

even stages◮ if ab not already in a block, add a new point n and a block abn

odd stages◮ for each pair ab consider the cycle graph G(a, b) (union of paths

and isolated vertices)◮ suppose x and y are end points (not same cycle) or isolated◮ if x is isolated or an a-edge, and y is isolated or a b-edge

add blocks bxn and any(swap colours if colours are reversed)

◮ if x and y are on a-edges add blocks bxn1, an1n2 and bn2y(swap colours if colours are reversed)

After countably many steps we have a countably INFINITE

perfect Steiner triple system (CAMERON, BSW 2012)

It is possible to construct 2ℵ0 non-isomorphic perfect systems this way

Route map

Constructions

Fraïssé’s TheoremSuppose C is a class of finitely generated structures such that

C is closed under isomorphisms

C contains only countably many members up to isomorphism

C has the Hereditary Property, HP

C has the Joint Embedding Property, JEP

C has the Amalgamation Property, AP

Then there is a countable homogeneous structure S

which is universal for C

unique up to isomorphisms

(FRAÏSSÉ 1954, JÓNSSON 1956)

We call S the Fraïssé limit of C

Such a class C is called an amalgamation class

Regard a Steiner triple system as a Steiner quasigroup

a ◦ b = c if and only if {a, b, c} is a block (and x ◦ x = x)

Then substructures (in the sense of model theory) are subsystems

The amalgamation property holds because any pair of systems with acommon subsystem can be completed to a finite system

The class of all finite Steiner triple systems is an amalgamation class— the Fraïssé limit is the unique (up to isomorphism)universal homogeneous locally finite Steiner triple system , U

(CAMERON 2007?)

Use k-uniform hypergraphs to build designs:

colour the points of the hypergraph to get s point orbits

each hyperedge has ni points of colour i

“orient” the hypergraph by distinguishing (t − k) points on eachhyperedge – called apex/apices

◮ each point is an apex of at most one hyperedge◮ two edges have at most t − 1 points in common

For a block transitive t-(v , k , 1) design with s point orbits, use ak-uniform hypergraph with s colours on the points

We defineδ(B) = |B| − (t − k)|EB |

We say A is “closed” in B if any apex of a hyperedge b ∈ Bis also in A , then b ∈ A

If δ(A) ≤ δ(B′) for all A ⊆ B′ ⊆ B then we can “orient” B so that A is“closed” in it

Apply Fraïssé’s Theorem to get a homogeneous structurethat is a block transitive t-(v , k , 1) design with s point orbits(s ≤ k/t )

(EVANS 2004, HRUSHOVSKI 1993)

Thank you!

[thanks go to Matt Tapp for help with the pictures]

Infinite designs -- a quick tourcamconf/talks/webb.pdfInﬁnite designs – a quick tour Bridget S...

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