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Another family of Steiner triple systems without almost parallel - - PowerPoint PPT Presentation
Another family of Steiner triple systems without almost parallel - - PowerPoint PPT Presentation
Another family of Steiner triple systems without almost parallel classes Daniel Horsley (Monash University) Joint work with Darryn Bryant (University of Queensland) Steiner triple systems Steiner triple systems Steiner triple systems Steiner
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
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Steiner triple systems
An STS(7)
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Steiner triple systems
An STS(7) Theorem (Kirkman 1847) An STS(v) exists if and only if v ≥ 1 and v ≡ 1 or 3 (mod 6).
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Parallel classes
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Parallel classes
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Parallel classes
An STS(9)
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Parallel classes
An STS(9)
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Parallel classes
An STS(9) with a PC
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Almost parallel classes
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Almost parallel classes
An STS(13)
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Almost parallel classes
An STS(13)
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Almost parallel classes
An STS(13) with an APC
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What’s known: small orders
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What’s known: small orders
◮ The unique STS(7) has no APC.
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What’s known: small orders
◮ The unique STS(7) has no APC. ◮ The unique STS(9) has a PC.
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What’s known: small orders
◮ The unique STS(7) has no APC. ◮ The unique STS(9) has a PC. ◮ Both STS(13)s have an APC.
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What’s known: small orders
◮ The unique STS(7) has no APC. ◮ The unique STS(9) has a PC. ◮ Both STS(13)s have an APC. ◮ All but 10 of the 80 STS(15)s have PCs.
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What’s known: small orders
◮ The unique STS(7) has no APC. ◮ The unique STS(9) has a PC. ◮ Both STS(13)s have an APC. ◮ All but 10 of the 80 STS(15)s have PCs. ◮ All but 2 of the 11, 084, 874, 829 STS(19)s have an APC. (Colbourn et al.)
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What’s known: small orders
◮ The unique STS(7) has no APC. ◮ The unique STS(9) has a PC. ◮ Both STS(13)s have an APC. ◮ All but 10 of the 80 STS(15)s have PCs. ◮ All but 2 of the 11, 084, 874, 829 STS(19)s have an APC. (Colbourn et al.) ◮ 12 STS(21)s are known to have no parallel class. (Mathon, Rosa)
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What’s guessed
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What’s guessed
Conjecture (Mathon, Rosa) There is an STS(v) with no PC for all v ≡ 3 (mod 6) except v = 3, 9. Conjecture (Rosa, Colbourn) There is an STS(v) with no APC for all v ≡ 1 (mod 6) except v = 13.
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What’s guessed
Conjecture (Mathon, Rosa) There is an STS(v) with no PC for all v ≡ 3 (mod 6) except v = 3, 9. Conjecture (Rosa, Colbourn) There is an STS(v) with no APC for all v ≡ 1 (mod 6) except v = 13. No example is known of an STS which does not contain a set of ⌊ v
3⌋ − 1
disjoint triples.
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What’s known: in general
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What’s known: in general
v ≡ 3 (mod 6)
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What’s known: in general
v ≡ 3 (mod 6) No infinite family of STSs of order 3 (mod 6) without PCs is known.
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What’s known: in general
v ≡ 3 (mod 6) No infinite family of STSs of order 3 (mod 6) without PCs is known. v ≡ 1 (mod 6)
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What’s known: in general
v ≡ 3 (mod 6) No infinite family of STSs of order 3 (mod 6) without PCs is known. v ≡ 1 (mod 6) Theorem (Wilson, 1992) For each odd n there is an STS(2n − 1) with no almost parallel class.
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What’s known: in general
v ≡ 3 (mod 6) No infinite family of STSs of order 3 (mod 6) without PCs is known. v ≡ 1 (mod 6) Theorem (Wilson, 1992) For each odd n there is an STS(2n − 1) with no almost parallel class.
Wilson’s examples are projective triple systems.
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What’s known: in general
v ≡ 3 (mod 6) No infinite family of STSs of order 3 (mod 6) without PCs is known. v ≡ 1 (mod 6) Theorem (Wilson, 1992) For each odd n there is an STS(2n − 1) with no almost parallel class.
Wilson’s examples are projective triple systems.
Theorem (Bryant, Horsley) For each n there is an STS(2(3n) + 1) with no almost parallel class.
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Affine triple systems
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Affine triple systems
◮ An STS(3n).
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3.
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
Note that:
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
Note that:
◮ Each triple adds to zero. ◮ The whole point set adds to zero.
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
Note that:
◮ Each triple adds to zero. ◮ The whole point set adds to zero.
1-dimensional subspaces have the form {0, a, 2a}.
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Affine triple systems
◮ An STS(3n). ◮ Points are Z3 × · · · × Z3. ◮ Triples are the 1-dimensional subspaces and their translates.
The affine STS(9):
0,0 0,1 0,2 1,0 1,1 1,2 2,0 2,1 2,2
Note that:
◮ Each triple adds to zero. ◮ The whole point set adds to zero.
1-dimensional subspaces have the form {0, a, 2a}. Translates of these have the form {b, a + b, 2a + b}.
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Our Proof
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0)
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0)
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC.
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
Adding the labels of the black vertices:
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
Adding the labels of the black vertices: 0 + · · · + 0
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
Adding the labels of the black vertices: 0 + · · · + 0 = 2
- x∈Z3×···×Z3
x
- − a − 2b
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
Adding the labels of the black vertices: 0 + · · · + 0 = 2
- x∈Z3×···×Z3
x
- − a − 2b
0 = 0 − a − 2b
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
Adding the labels of the black vertices: 0 + · · · + 0 = 2
- x∈Z3×···×Z3
x
- − a − 2b
0 = 0 − a − 2b a = b
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Our Proof
Affine STS(3n)
(2, . . . , 2) (0, . . . , 0) (2, . . . , 2) (0, . . . , 0) ∞
STS(2(3n) + 1)
Suppose this STS has an APC. The missing vertex must be in the subsystem.
a
So one triple in the APC contains ∞.
b b
Adding the labels of the black vertices: 0 + · · · + 0 = 2
- x∈Z3×···×Z3
x
- − a − 2b
0 = 0 − a − 2b a = b Contradiction.
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Our Results
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Our Results
Theorem For each n there is an STS(2(3n) + 1) with no almost parallel class.
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Our Results
Theorem For each n there is an STS(2(3n) + 1) with no almost parallel class. Theorem If q, n and d are integers with 1 ≤ d ≤ n such that both q and qd − 1 are prime powers, then there exists a (qn(qd − 1) + 1, qd, n−1
d−1
- q)-design
with no almost parallel class.
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