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Trails of triples in Steiner triple systems
Daniel Horsley (Monash University, Australia)
Joint work with Charles Colbourn and Chengmin Wang
Trails of triples in Steiner triple systems Daniel Horsley (Monash - - PowerPoint PPT Presentation
Trails of triples in Steiner triple systems Daniel Horsley (Monash - - PowerPoint PPT Presentation
Trails of triples in Steiner triple systems Daniel Horsley (Monash University, Australia) Joint work with Charles Colbourn and Chengmin Wang Steiner triple systems and block colourings Steiner triple systems and block colourings 1 2 3 1 9
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Steiner triple systems and block colourings
1 2 3 4 5 6 7 8 9
1 2 3 1 4 7 1 5 9 1 6 8 2 4 9 2 5 8 2 6 7 3 4 8 3 5 7 3 6 9 4 5 6 7 8 9
An STS(9)
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Steiner triple systems and block colourings
1 2 3 4 5 6 7 8 9
1 2 3 1 4 7 1 5 9 1 6 8 2 4 9 2 5 8 2 6 7 3 4 8 3 5 7 3 6 9 4 5 6 7 8 9
An STS(9) admitting a colouring of type (3, 3, 2, 2, 1, 1)
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Theorem [Kirkman (1847)] An STS(v) exists if and only if v ≥ 1 and v ≡ 1, 3 (mod 6).
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Partial Steiner triple systems and block colourings
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Partial Steiner triple systems and block colourings
1 2 3 4 5 6 7 8
1 2 4 2 3 5 3 4 6 4 5 7 5 6 8 1 7 8 2 6 7
A PSTS(8)
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Partial Steiner triple systems and block colourings
1 2 3 4 5 6 7 8
1 2 4 2 3 5 3 4 6 4 5 7 5 6 8 1 7 8 2 6 7
A PSTS(8) admitting a colouring of type (2, 2, 1, 1, 1)
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Necessary conditions and a conjecture
For a PSTS(v) which admits a colouring of type (c1, c2, . . . , ct) to exist we must have (i) ci ≤ ⌊ v
3⌋ for i = 1, 2, . . . , t; and
(ii) c1 + c2 + · · · + ct ≤ µ(v), where µ(v) is the maximum number of triples in a PSTS(v).
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Necessary conditions and a conjecture
For a PSTS(v) which admits a colouring of type (c1, c2, . . . , ct) to exist we must have (i) ci ≤ ⌊ v
3⌋ for i = 1, 2, . . . , t; and
(ii) c1 + c2 + · · · + ct ≤ µ(v), where µ(v) is the maximum number of triples in a PSTS(v). If a colour type (c1, c2, . . . , ct) satisfies (i) and (ii) then we will say it is v-feasible.
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Necessary conditions and a conjecture
For a PSTS(v) which admits a colouring of type (c1, c2, . . . , ct) to exist we must have (i) ci ≤ ⌊ v
3⌋ for i = 1, 2, . . . , t; and
(ii) c1 + c2 + · · · + ct ≤ µ(v), where µ(v) is the maximum number of triples in a PSTS(v). If a colour type (c1, c2, . . . , ct) satisfies (i) and (ii) then we will say it is v-feasible. Conjecture [Colbourn, Horsley, Wang (2011)] Let v ≥ 14. For every v-feasible colour type (c1, c2, . . . , ct) there exists a PSTS(v) admitting a colouring of type (c1, c2, . . . , ct).
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Necessary conditions and a conjecture
For a PSTS(v) which admits a colouring of type (c1, c2, . . . , ct) to exist we must have (i) ci ≤ ⌊ v
3⌋ for i = 1, 2, . . . , t; and
(ii) c1 + c2 + · · · + ct ≤ µ(v), where µ(v) is the maximum number of triples in a PSTS(v). If a colour type (c1, c2, . . . , ct) satisfies (i) and (ii) then we will say it is v-feasible. Conjecture [Colbourn, Horsley, Wang (2011)] Let v ≥ 14. For every v-feasible colour type (c1, c2, . . . , ct) there exists a PSTS(v) admitting a colouring of type (c1, c2, . . . , ct).
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Reasons to care
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Reasons to care
◮ A Kirkman triple system is an STS(6t + 3) admitting a colouring of type
(2t + 1, 2t + 1, . . . , 2t + 1).
◮ A nearly Kirkman triple system is a maximum PSTS(6t) admitting a
colouring of type (2t, 2t, . . . , 2t).
◮ A Hanani triple system is an STS(6t + 1) admitting a colouring of type
(2t, 2t, . . . , 2t, t).
◮ The conjecture is also related to 3-frames and to many other block
colouring problems.
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Reasons to care
◮ A Kirkman triple system is an STS(6t + 3) admitting a colouring of type
(2t + 1, 2t + 1, . . . , 2t + 1).
◮ A nearly Kirkman triple system is a maximum PSTS(6t) admitting a
colouring of type (2t, 2t, . . . , 2t).
◮ A Hanani triple system is an STS(6t + 1) admitting a colouring of type
(2t, 2t, . . . , 2t, t).
◮ The conjecture is also related to 3-frames and to many other block
colouring problems.
◮ A strong Kirkman signal set SKSS(v, m) is a maximum PSTS(v)
admitting a colouring of type (m, m, . . . , m, r), where 1 ≤ r ≤ m.
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Good news and bad news
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Good news and bad news
We’ll say that a colour type is v-realisable if there exists a PSTS(v) which admits that colour type.
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Good news and bad news
We’ll say that a colour type is v-realisable if there exists a PSTS(v) which admits that colour type. The realisability of many colour types follows immediately from the realisability
- f others.
For example, any STS(15) admitting a colouring of type (5, 5, 5, 5, 5, 5, 5) must also admit a colouring of type (5, 5, 5, 5, 5, 4, 3, 2, 1).
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Good news and bad news
We’ll say that a colour type is v-realisable if there exists a PSTS(v) which admits that colour type. The realisability of many colour types follows immediately from the realisability
- f others.
For example, any STS(15) admitting a colouring of type (5, 5, 5, 5, 5, 5, 5) must also admit a colouring of type (5, 5, 5, 5, 5, 4, 3, 2, 1). But, for any large v, there are still vast numbers of feasible colour types which are not implied in this way. For example, for v = 15, (4, 4, 4, 4, 4, 4, 4, 4, 3) is not implied by (5, 5, 5, 5, 5, 5, 5) (or any other colour type).
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Trails of triples
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t. Such an ordering of the triples of a PSTS is called m-pessimal.
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Trails of triples
If we can order the triples of a PSTS in such a way that any m consecutive triples are vertex disjoint, then the PSTS must admit all colourings of type (c1, c2, . . . , ct) where ci ≤ m for i = 1, 2, . . . , t. Such an ordering of the triples of a PSTS is called m-pessimal. Note that m can be at most ⌊ v
3⌋.
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Trails of triples
Theorem [Cohen, Colbourn (2000)] For each v ≥ 1 such that v ≡ 1, 3 (mod 6), there is an STS(v) admitting an ⌊ v+6
9 ⌋-pessimal ordering.
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Trails of triples
Theorem [Cohen, Colbourn (2000)] For each v ≥ 1 such that v ≡ 1, 3 (mod 6), there is an STS(v) admitting an ⌊ v+6
9 ⌋-pessimal ordering.
Theorem [Colbourn, Horsley, Wang (2011)] For each sufficiently large v, there is a maximum PSTS(v) admitting an m-pessimal ordering where m = v
3(1 − o(1)).
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Trails of triples
Theorem [Cohen, Colbourn (2000)] For each v ≥ 1 such that v ≡ 1, 3 (mod 6), there is an STS(v) admitting an ⌊ v+6
9 ⌋-pessimal ordering.
Theorem [Colbourn, Horsley, Wang (2011)] For each sufficiently large v, there is a maximum PSTS(v) admitting an m-pessimal ordering where m = v
3(1 − o(1)).
In fact m = ⌊ 1
3(v − (9v)2/3)⌋ + O(v 1/3).
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Trails of triples
Theorem [Cohen, Colbourn (2000)] For each v ≥ 1 such that v ≡ 1, 3 (mod 6), there is an STS(v) admitting an ⌊ v+6
9 ⌋-pessimal ordering.
Theorem [Colbourn, Horsley, Wang (2011)] For each sufficiently large v, there is a maximum PSTS(v) admitting an m-pessimal ordering where m = v
3(1 − o(1)).
In fact m = ⌊ 1
3(v − (9v)2/3)⌋ + O(v 1/3).
Corollary For each sufficiently large v, each v-feasible colour type (c1, c2, . . . , ct) with ci ≤ m for i = 1, 2, . . . , t is realisable.
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Proof sketch
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Proof sketch
Lemma Let n be an odd integer. There exists an decomposition of the complete tripartite graph Kn,n,n into triples which admits an (n − 2)-pessimal
- rdering.
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Proof of lemma
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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Example: an STS(73)
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General case
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General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
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General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
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Theorem [Colbourn, Horsley, Wang (2011)] For each sufficiently large v, there is a maximum PSTS(v) admitting an m-pessimal ordering where m = v
3(1 − o(1)).
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Future directions
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Future directions
◮ Can we improve our result to m = v
3 − O(1)?
◮ Can we make this construction better by making it recursive? ◮ Prove the colouring conjecture. ◮ Latin square equivalents of these problems. ◮ For sufficiently large v, is there a maximum PSTS(v) that admits all
v-feasible colourings?
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