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Properties of the Steiner Triple Systems of Order 19

2010
*
Electronic Journal of Combinatorics
*

*Properties*

*of*

*the*11$\,$084$\,$874$\,$829

*Steiner*

*triple*

*systems*

*of*

*order*

*19*are examined. ... In particular, there is exactly one 5-sparse, but no 6-sparse, STS(

*19*); there is exactly one uniform STS(

*19*); there are exactly two STS(

*19*) with no almost parallel classes; all STS(

*19*) have chromatic number ... Kiviluoto, A catalogue

*of*

*the*

*Steiner*

*triple*

*systems*

*of*

*order*

*19*, Bull. Inst. Combin. ...

##
###
The Steiner quadruple systems of order 16

2006
*
Journal of combinatorial theory. Series A
*

*Properties*

*of*

*the*designs-including

*the*

*orders*

*of*

*the*automorphism groups and

*the*structures

*of*

*the*derived

*Steiner*

*triple*

*systems*

*of*

*order*15-are tabulated. ...

*The*

*Steiner*quadruple

*systems*

*of*

*order*16 are classified up to isomorphism by means

*of*an exhaustive computer search.

*The*number

*of*isomorphism classes

*of*such designs is 1,054,163. ... Acknowledgments

*The*authors are grateful to Juhani Markula and Jukka Seppänen at

*the*Computing Centre

*of*Helsinki University

*of*Technology for providing computing resources. ...

##
###
Steiner Triple Systems with High Chromatic Index

2017
*
SIAM Journal on Discrete Mathematics
*

It is conjectured that every

doi:10.1137/17m1114338
fatcat:lcvt2pl7xbgmhbboiwjbtcbk2m
*Steiner**triple**system**of**order*v ≠ 7 has chromatic index at most (v+3)/2 when v ≡ 3 6 and at most (v+5)/2 when v ≡ 1 6. ... Herein, we construct a*Steiner**triple**system**of**order*v with chromatic index at least (v+3)/2 for each integer v ≡ 3 6 such that v ≥ 15, with four possible exceptions. ... Herke for writing*the*code that generates random*Steiner**triple**systems*. ...##
###
Page 23 of Mathematical Reviews Vol. , Issue 82a
[page]

1982
*
Mathematical Reviews
*

*The*authors prove that any partial

*Steiner*

*triple*

*system*

*of*

*order*n can be embedded in a

*Steiner*

*triple*

*system*

*of*

*order*v, for any v=1 or 3 (mod6) with v > 4n. ...

*The*authors prove that if there exists a partial

*Steiner*

*triple*

*system*

*of*

*order*n con- taining y

*triples*, then there exists an equitable partial

*Steiner*

*triple*

*system*

*of*

*order*n containing y

*triples*. ...

##
###
Embedding Steiner triple systems in hexagon triple systems

2009
*
Discrete Mathematics
*

*The*inside

*triples*form a partial

*Steiner*

*triple*

*system*. We show that any

*Steiner*

*triple*

*system*

*of*

*order*n can be embedded in

*the*inside

*triples*

*of*a hexagon

*triple*

*system*

*of*

*order*approximately 3n. ... A hexagon

*triple*

*system*

*of*

*order*n is a pair (X, H ) where H is a collection

*of*edge disjoint hexagon

*triples*which partitions

*the*edge set

*of*K n with vertex set X . ... If a

*Steiner*

*triple*

*system*

*of*

*order*n can be embedded in a hexagon

*triple*

*system*

*of*

*order*v, then every

*Steiner*

*triple*

*system*

*of*

*order*n can be embedded in a hexagon

*triple*

*system*

*of*

*order*v. ...

##
###
Page 1808 of Mathematical Reviews Vol. , Issue 85e
[page]

1985
*
Mathematical Reviews
*

known about

*Steiner**triple**systems**of*small*orders*(up to 15): listing*of**the**systems*themselves, together with*the*various*properties*they possess. ... No one who does research on*Steiner**triple**systems*(or more generally block designs) should be without this paper. It is*the*dictionary*of**Steiner**triple**systems**of*small*orders*. ...##
###
A second infinite family of Steiner triple systems without almost parallel classes

2013
*
Journal of combinatorial theory. Series A
*

*The*only previously known examples

*of*

*Steiner*

*triple*

*systems*

*of*

*order*congruent to 1 (mod 6) without almost parallel classes were

*the*projective

*triple*

*systems*

*of*

*order*2 n − 1 with n odd, and 2

*of*

*the*... 11, 084, 874, 829

*Steiner*

*triple*

*systems*

*of*

*order*

*19*. called blocks, such that each (unordered) pair

*of*points occurs in exactly λ blocks in B. ... Acknowledgements:

*The*authors acknowledge

*the*support

*of*

*the*Australian Research Council via grants DE120100040, DP120100790 and DP120103067. ...

##
###
Page 5427 of Mathematical Reviews Vol. , Issue 87j
[page]

1987
*
Mathematical Reviews
*

R. (4-HWAT) 87j:05039

*Steiner**triple**systems**of**order**19*associated with a certain type*of*projective plane*of**order*10. Period. Math. Hungar. 17 (1986), no. 3, 177-184. ... From this he cleverly constructs a*Steiner**triple**system*(STS) on*19*points with three rather restrictive*properties*.*The*author has found no such STS among*the*many examples he has checked. ...##
###
Steiner Triple Systems and Existentially Closed Graphs

2005
*
Electronic Journal of Combinatorics
*

We investigate

doi:10.37236/1939
fatcat:fiw7mrm74rc67l7tbzfqv2p7ra
*the*conditions under which a*Steiner**triple**system*can have a 2- or 3-existentially closed block intersection graph. ... Acknowledgement*The*authors would like to thank Petteri Kaski and PatricÖstergård, Helsinki University*of*Technology, for making available to us listings*of**Steiner**triple**systems**of**order**19*. ... This characterization implies that such graphs can exist only for a limited range*of**orders**of**Steiner**triple**system*, possibly even just one*order*;*19*points. ...##
###
On 2-ranks of Steiner triple systems

1995
*
Electronic Journal of Combinatorics
*

$-rank from

doi:10.37236/1203
fatcat:d7ti2g23djge3pbstxj3t5f6qe
*Steiner**triple**systems**of*a specified smaller*order*. ... When*the**Steiner**triple**system*is*of*2-rank less than*the*number*of*points*of**the**system*,*the*carrier organizes all*the*information necessary to construct directly all*systems**of**the*given*order*and $2 ... In this case*of**Steiner**triple**systems**of**order*8 (on*19*points)*the*2-rank is either 18 or*19*(from Theorem 3.4) and Brendan McKay estimates that*the*number*of**Steiner**triple**systems*is 11 or 12 billion ...##
###
Countable homogeneous Steiner triple systems avoiding specified subsystems
[article]

2020
*
arXiv
*
pre-print

In this article we construct uncountably many new homogeneous locally finite

arXiv:2006.04605v1
fatcat:wtyf7rs3uva53pqftdta6boaiu
*Steiner**triple**systems**of*countably infinite*order*as Fraïssé limits*of*classes*of*finite*Steiner**triple**systems*avoiding certain ... subsystems*of**the*original partial*system*. ... Webb wishes to acknowledge*the*support*of*a 2014 Ethel Raybould Visiting Fellowship from*the*School*of*Physical Sciences,*The*University*of*Queensland. ...##
###
Optimal and Pessimal Orderings of Steiner Triple Systems in Disk Arrays
[chapter]

2000
*
Lecture Notes in Computer Science
*

*Steiner*

*triple*

*systems*are well studied combinatorial designs that have been shown to possess

*properties*desirable for

*the*construction

*of*multiple erasure codes in RAID architectures. ...

*The*

*ordering*

*of*

*the*columns in

*the*parity check matrices

*of*these codes a ects

*system*performance. ... Acknowledgements Research

*of*

*the*authors is supported by

*the*Army Research O ce (USA) under Grant no. DAAG55-98-1-0272 (Colbourn). ...

##
###
Optimal and pessimal orderings of Steiner triple systems in disk arrays

2003
*
Theoretical Computer Science
*

*Steiner*

*triple*

*systems*are well studied combinatorial designs that have been shown to possess

*properties*desirable for

*the*construction

*of*multiple erasure codes in RAID architectures. ...

*The*

*ordering*

*of*

*the*columns in

*the*parity check matrices

*of*these codes a ects

*system*performance. ... Acknowledgements Research

*of*

*the*authors is supported by

*the*Army Research O ce (USA) under Grant no. DAAG55-98-1-0272 (Colbourn). ...

##
###
The centre of a sloop

1990
*
The Australasian Journal of Combinatorics
*

*The*3-subsets are called

*the*blocks

*of*S.

*The*

*order*

*of*

*the*

*Steiner*

*triple*

*system*is v,

*the*size

*of*

*the*set S. ... A

*Steiner*

*triple*

*system*n is said t.o be a subsystem

*of*S if, R ~ S and

*the*blocks

*of*n are also blocks

*of*S. ...

*The*software mentioned in this paper was written by Martin Sharry. I wish to acknowledge

*the*support

*of*a Special Projects Grant from

*the*University

*of*Queensland. -'EFERENCES. ...

##
###
Page 8483 of Mathematical Reviews Vol. , Issue 2004k
[page]

2004
*
Mathematical Reviews
*

Summary: “

*The*automorphism group*of**the**Steiner**triple**system**of**order*v = 6s +3, obtained from*the*Bose construction using any abelian group G*of**order*2s +1, is determined. ... [Rosa, Alexander] (3-MMAS-MS; Hamilton, ON) Specialized block-colourings*of**Steiner**triple**systems*and*the*upper chromatic index. (English summary) Graphs Combin.*19*(2003), no. 3, 335-345. ...
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