We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2ℵ0 non-isomorphic perfect systems

B. S. Webb

How To Cite

P. J. Cameron

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Cameron, Peter Jephson

Webb, Bridget

University of St. Andrews - Pure

Perfect countably infinite Steiner triple systems

We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2 ℵ0 non-isomorphic perfect systems

P J Cameron

B S Webb

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AUSTRALASIAN JOURNAL OF COMBINATORICS
Volume 54 (2012), Pages 273–278
Perfect countably infinite Steiner triple systems
P. J. Cameron
School of Mathematical Sciences
Queen Mary, University of London
Mile End Road, London, E1 4NS
U.K.
p.j.cameron@qmul.ac.uk
B. S. Webb
Department of Mathematics and Statistics
The Open University
Walton Hall, Milton Keynes, MK7 6AA
U.K.
b.s.webb@open.ac.uk
Abstract
We use a free construction to prove the existence of perfect Steiner triple
systems on a countably infinite point set. We use a specific countably
infinite family of partial Steiner triple systems to start the construction,
thus yielding 2ℵ0 non-isomorphic perfect systems.
1 Introduction
The problem of finding perfect finite Steiner triple systems is difficult; only fourteen
such systems are known [7, 8]. The question of the existence of perfect countably
infinite Steiner triple systems was posed at the British Combinatorial Conference
in 2009 [3]. In this paper we show that it is possible to use a free construction
to obtain a perfect countably infinite Steiner triple system, and furthermore, by
carefully choosing the partial Steiner triple system used to start the construction,
2ℵ0 non-isomorphic perfect countably infinite Steiner triple systems can be produced.
We begin with some basic definitions and an explanation of the terminology.
A finite Steiner triple system of order v is a pair (V,B) where V is a finite set
of v elements (the points) and B is a collection of 3-element subsets (the blocks) of
V such that each 2-element subset of V is contained in exactly one block of B. It is
well known that a Steiner triple system (STS) of order v exists if and only if v ≡ 1
or 3 (mod 6); such values of v are called admissible. If we replace the requirement
that V be finite by the requirement that V have cardinality ℵ0, then the resulting
pair (V,B) is called a countably infinite Steiner triple system.
274 P.J. CAMERON AND B.S. WEBB
A partial Steiner triple system (or configuration) is a set of 3-element subsets
(blocks), taken from a point set V , which has the property that every pair of distinct
elements of V occurs in at most one block. A partial Steiner triple system (partial
STS) is called finite if it has a finite set of blocks; it is called countably infinite if the
block set is countably infinite. If C is a partial STS, we denote its set of points by
P (C). The degree of a point in a partial STS is the cardinality of the set of blocks
that contain that point. A partial STS is connected if, for any pair of its points x
and y, there is a sequence x = x0, x1, . . . xn = y of its points xi, such that the pair
xi and xi+1 are contained in together in a block, for all i.
A sub-system (V ′,B′) of an STS (V,B) is a Steiner triple system with V ′ ⊆ V
and B′ ⊆ B. A partial sub-system of an STS, or a partial STS, is defined analogously
except that each 2-element subset of V ′ is contained in at most one block of B′ rather
than in exactly one.
For any two points a and b of a (finite or countably infinite) Steiner triple system
S, contained in a block abc, the cycle graph G(a, b) has vertex set V \ {a, b, c}, with
an edge coloured a (resp., b) joining x to y if and only if axy is a block (resp.,
bxy is a block). In the finite case, it is well known that G(a, b) is a set of disjoint
cycles {Cn1 , Cn2 , . . . , Cnr}, where n1 + n2 + . . . + nr = v − 3; hence the name cycle
graph. Moreover each ni is even with ni ≥ 4. In the infinite case, besides finite
cycles, G(a, b) may have “infinite cycles”, components that we interpret as “two-way
infinite paths”. We define the cycle graph, G(a, b), of any two points a and b of a
partial STS analogously, noting that here a and b may not be contained together in
a block.
A uniform STS has each cycle graph G(a, b) isomorphic and a perfect STS has
each cycle graph G(a, b) a single cycle (in analogy with a perfect 1-factorisation where
the union of the 1-factors is a Hamiltonian cycle). Thus, a perfect STS is uniform,
but the converse is not true. The existence of uncountably many uniform countably
infinite Steiner triple systems is known [4], but none of these systems is perfect.
Note that an infinite perfect STS cannot be larger than countable since a Hamil-
tonian cycle cannot be larger than countable.
2 Results
Many countably infinite structures can be constructed “freely” with any special prop-
erties “built in” during the construction. Examples include Steiner systems with large
values of t, and highly transitive Steiner systems [2]. Here we use such a construction
to obtain a perfect countably infinite Steiner triple system by starting with a suitable
partial STS and carefully “building in” the perfect property during the construction.
The following are the steps required in a free construction of a countably infinite
Steiner triple system from any finite (or countably infinite) partial STS.
• Start with a finite (or countably infinite) partial Steiner triple system
• Adjoin alternatively:
– new blocks incident with those pairs of points not already in a block
INFINITE STEINER TRIPLE SYSTEMS 275
– new points so each existing block has 3 points
• After countably many steps the result is a Steiner triple system.
To obtain a perfect countably infinite Steiner triple system we must ensure that
no cycle graph G(a, b) in our system contains a finite cycle, and that each cycle graph
G(a, b) is connected. Thus we start with a partial STS with no cycle graph G(a, b)
containing a finite cycle. At each stage of the construction we insist that any new
block contains at least one new point. This rules out any finite cycle being created in
any cycle graph G(a, b) because if the block axy creates a cycle in G(a, b) then there
exists an xy-path in G(a, b) say [x, w, v, ..., z, y] where possibly v = z. But then we
have blocks avw, bwx, byz; so all of a, x, y are existing points, and axy cannot be a
new block. In this way, all that we need to check is that the result is a countably
infinite Steiner triple system and that all the cycle graphs G(a, b) are connected.
Theorem 2.1 There exists a countably infinite perfect Steiner triple system.
Proof We start with a (finite or countably infinite) partial STS with no cycle graph
G(a, b) containing a finite cycle and we construct a countably infinite perfect Steiner
triple system in stages.
At even-numbered stages, we list all pairs of existing points, and for each such
pair ab which is not already contained in a block, we add a new point n and a block
abn.
At odd-numbered stages, we list all pairs of existing points. For each such pair ab,
consider the cycle graph G(a, b). This will be a union of paths and isolated vertices.
Suppose that x and y are either ends of paths or isolated vertices (if both are path
ends, it should not be the same path). Now add new vertices and blocks as follows:
• If x is isolated or on an a-coloured edge and y is isolated or on a b-coloured
edge, add blocks bxn and any, where n is a new point. Similarly if the colours
are reversed.
• If x and y are on a-coloured edges, then add blocks bxn1, an1n2, and bn2y.
Similarly if they are both on b-coloured edges.
An easy check shows that we never introduce a second block containing two points
already in a block. (In the first case, for example, if there were a block containing b
and x, then x would not be isolated or the end of a path in G(a, b).)
Now given any two points, there will be a first stage of the construction at which
they have both appeared; and in the next even-numbered stage, a block containing
them will be introduced. So we construct a Steiner triple system S.
Given points a and b, and x and y in V \ {a, b, c}, where abc is a block, at the
next odd-numbered stage after all these points have been introduced, x and y will
be in the same connected component of the cycle graph G(a, b). So the countably
infinite Steiner triple system produced is perfect.
To construct non-isomophic countably infinite perfect Steiner triple systems in
this way we need to choose the partial STS used at the start of the construction
276 P.J. CAMERON AND B.S. WEBB
carefully. The infinite family of partial STS used to construct the uniform count-
ably infinite Steiner triple systems in [4] are ‘perfect’ for our needs. We take the
construction from the proof of Lemma 2.3 in that paper.
Let D1 be a partial STS with P (D1) = {a, b, c, d, e, f, g, h, i} and the six blocks:
{a, b, c}, {d, e, f}, {g, h, i}, {a, d, g}, {b, e, h}, {c, f, i}.
It is connected and has 9 points all of degree 2.
We form an infinite sequence of partial STS called D1,D2,D3 . . . starting from
D1. We construct Di+1 from Di in the following manner. Di is a connected partial
STS with 3(i + 2) points all of degree 2, where i ≥ 1. There exist two points of
P (Di), say x, y, that do not lie together in a block of Di. Replace x by a new point
x1 in one of the two blocks containing x, and replace x by another new point x2 in
the other block containing x. Carry out a similar replacement of y by new points y1
and y2. This may disconnect the configuration, but we can assume that x1 remains
in the same component as y1. Now introduce a new point z and two new blocks
x1y2z and x2y1z. The resulting partial STS Di+1 has 3(i + 3) points, all of degree 2,
and is connected. Clearly no Di can contain any other Dj as a (partial) sub-system.
We consider the cycle graphs G(a, b) of Di; because the degree of each point
is 2, there are at most 4 edges in each such graph (when the points a and b are
not contained together in a block). The shortest cycle possible in any cycle graph
G(a, b) is of length 4 and occurs only when the STS or partial STS contains a Pasch
configuration (also called a quadrilateral) with four blocks of the form
{a, x, y}, {b, y, z}, {a, z, t}, {b, x, t}.
The way that the partial STS Di are constructed precludes this possibility, so none
of the cycle graphs G(a, b) of any of the Di contains a cycle.
Let D0 be the trivial partial STS comprising just a single point and no blocks.
Denote by C the partial STS formed from some collection of partial STS Di (i ≥ 0);
note that they need not all be distinct.
Theorem 2.2 There exists a countably infinite perfect Steiner triple system contain-
ing the partial STS C, which contains no other connected finite partial sub-system all
of whose points have degree 2.
Proof The cycle graphs G(a, b) of the partial STS C contain no cycles since each
component of this partial STS is simply one of the Di and there are no edges in any
G(a, b) between these components.
We therefore follow the construction in stages, as in the proof of Theorem 2.1,
starting with the partial STS C. The resulting countably infinite Steiner triple system
will be perfect by Theorem 2.1.
At every stage in the construction, every new block added contains a new point,
and so no new finite partial sub-system can be formed all of whose points are of
degree 2, since there must be at least one new point of degree 1 on every block
added.
INFINITE STEINER TRIPLE SYSTEMS 277
Corollary 2.2.1 There are 2ℵ0 non-isomorphic countably infinite perfect Steiner
triple systems.
Proof If two distinct infinite subsets of {D1,D2,D3, . . .} are taken to form two
initial partial STS used in two applications of the construction, then the two result-
ing perfect countably infinite STS are non-isomorphic since they contain different
numbers of the partial STS Di. There are 2ℵ0 such subsets, so there are that many
non-isomorphic countably infinite STS.
3 Concluding Remarks
The existence of these perfect countably infinite Steiner triple systems fits with the
assertion that the question of the existence of countably infinite Steiner systems is in
general inordinately simpler than the finite case, since there is usually “space” to fit
everything in. Ad hoc methods, “free” constructions and more general constructions
using Model Theoretic results, such as the Fräıssé Limit can be used to construct
many countably infinite Steiner systems with specified properties.
Care has to be exercised though, as some infinite systems with specified properties
do not exist. On one hand, the existence of some systems is easily ruled out by
combinatorial restrictions; for example, the non-existence of a block-transitive, point-
intransitive Steiner triple system [1, 9], although block-transitive, point-intransitive
Steiner systems with block size greater than 3 do exist [6]. On the other hand, some
countably infinite Steiner systems with specified properties do not exist and there is
no obvious combinatorial reason why. It has recently been shown that all infinite
Steiner systems with block size strictly less than v are resolvable [5], and so no non-
resolvable infinite Steiner triple system exists. Note, however, that the projective
plane is an example of a non-resolvable Steiner system with block size equal to v.
Acknowledgments
This work was undertaken during the 35th ACCMCC held at Monash University,
where both authors were invited speakers.
References
[1] P. J. Cameron, An orbit theorem for Steiner triple systems, Discrete Math. 125
(1994), 97–100.
[2] P. J. Cameron, Infinite versions of some topics in finite geometry, “Geometrical
combinatorics,” Research Notes in Mathematics Series 114 (Eds.) F. C. Holroyd
and R. J. Wilson, Pitman, Boston, 1984, pp. 13–20.
[3] P. J. Cameron (ed.), Research problems from the BCC22, Discrete Math. 311
(2011), 1074–1083. Problem 506 (BCC22.2) proposed by M. J. Grannell, T. S.
Griggs and B. S. Webb.
278 P.J. CAMERON AND B.S. WEBB
[4] K. M. Chicot, M. J. Grannell, T. S. Griggs and B. S. Webb, On sparse countably
infinite Steiner triple systems, J. Combin. Des. 18 (2010), 115–122.
[5] P. Danziger, D. Horsley and B. S. Webb, Resolvability of Infinite Designs
preprint.
[6] D. M. Evans, Block transitive Steiner systems with more than one block orbit,
J. Combin. Des. 12 (2004), 459–465.
[7] A. D. Forbes, M. J. Grannell and T. S. Griggs, On 6-sparse Steiner triple sys-
tems, J. Combin. Theory Ser. A 114 (2007), 235–252.
[8] M. J. Grannell, T. S. Griggs and J. P. Murphy, Some new perfect Steiner triple
systems, J. Combin. Des. 7 (1999), 327–330.
[9] B. S. Webb, Orbits of infinite block designs, Discrete Math. 169 (1997), 287–292.
(Received 4 Feb 2012)


Cameron, Peter

Open Research Online

Open Research OnlineThe Open University’s repository of research publicationsand other research outputsPerfect countably infinite Steiner triple systemsJournal ItemHow to cite:Cameron, Peter and Webb, Bridget (2012). Perfect countably infinite Steiner triple systems. AustralasianJournal of Combinatorics, 54 pp. 273–278.For guidance on citations see FAQs.c© 2012 Australasian Journal of CombinatoricsVersion: Version of RecordLink(s) to article on publisher’s website:http://ajc.maths.uq.edu.auCopyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyrightowners. For more information on Open Research Online’s data policy on reuse of materials please consult the policiespage.oro.open.ac.ukAUSTRALASIAN JOURNAL OF COMBINATORICSVolume 54 (2012), Pages 273–278Perfect countably infinite Steiner triple systemsP. J. CameronSchool of Mathematical SciencesQueen Mary, University of LondonMile End Road, London, E1 4NSU.K.p.j.cameron@qmul.ac.ukB. S. WebbDepartment of Mathematics and StatisticsThe Open UniversityWalton Hall, Milton Keynes, MK7 6AAU.K.b.s.webb@open.ac.ukAbstractWe use a free construction to prove the existence of perfect Steiner triplesystems on a countably inﬁnite point set. We use a speciﬁc countablyinﬁnite family of partial Steiner triple systems to start the construction,thus yielding 2ℵ0 non-isomorphic perfect systems.1 IntroductionThe problem of ﬁnding perfect ﬁnite Steiner triple systems is diﬃcult; only fourteensuch systems are known [7, 8]. The question of the existence of perfect countablyinﬁnite Steiner triple systems was posed at the British Combinatorial Conferencein 2009 [3]. In this paper we show that it is possible to use a free constructionto obtain a perfect countably inﬁnite Steiner triple system, and furthermore, bycarefully choosing the partial Steiner triple system used to start the construction,2ℵ0 non-isomorphic perfect countably inﬁnite Steiner triple systems can be produced.We begin with some basic deﬁnitions and an explanation of the terminology.A finite Steiner triple system of order v is a pair (V,B) where V is a ﬁnite setof v elements (the points) and B is a collection of 3-element subsets (the blocks) ofV such that each 2-element subset of V is contained in exactly one block of B. It iswell known that a Steiner triple system (STS) of order v exists if and only if v ≡ 1or 3 (mod 6); such values of v are called admissible. If we replace the requirementthat V be ﬁnite by the requirement that V have cardinality ℵ0, then the resultingpair (V,B) is called a countably infinite Steiner triple system.274 P.J. CAMERON AND B.S. WEBBA partial Steiner triple system (or configuration) is a set of 3-element subsets(blocks), taken from a point set V , which has the property that every pair of distinctelements of V occurs in at most one block. A partial Steiner triple system (partialSTS) is called finite if it has a ﬁnite set of blocks; it is called countably infinite if theblock set is countably inﬁnite. If C is a partial STS, we denote its set of points byP (C). The degree of a point in a partial STS is the cardinality of the set of blocksthat contain that point. A partial STS is connected if, for any pair of its points xand y, there is a sequence x = x0, x1, . . . xn = y of its points xi, such that the pairxi and xi+1 are contained in together in a block, for all i.A sub-system (V ′,B′) of an STS (V,B) is a Steiner triple system with V ′ ⊆ Vand B′ ⊆ B. A partial sub-system of an STS, or a partial STS, is deﬁned analogouslyexcept that each 2-element subset of V ′ is contained in at most one block of B′ ratherthan in exactly one.For any two points a and b of a (ﬁnite or countably inﬁnite) Steiner triple systemS, contained in a block abc, the cycle graph G(a, b) has vertex set V \ {a, b, c}, withan edge coloured a (resp., b) joining x to y if and only if axy is a block (resp.,bxy is a block). In the ﬁnite case, it is well known that G(a, b) is a set of disjointcycles {Cn1 , Cn2 , . . . , Cnr}, where n1 + n2 + . . . + nr = v − 3; hence the name cyclegraph. Moreover each ni is even with ni ≥ 4. In the inﬁnite case, besides ﬁnitecycles, G(a, b) may have “inﬁnite cycles”, components that we interpret as “two-wayinﬁnite paths”. We deﬁne the cycle graph, G(a, b), of any two points a and b of apartial STS analogously, noting that here a and b may not be contained together ina block.A uniform STS has each cycle graph G(a, b) isomorphic and a perfect STS haseach cycle graph G(a, b) a single cycle (in analogy with a perfect 1-factorisation wherethe union of the 1-factors is a Hamiltonian cycle). Thus, a perfect STS is uniform,but the converse is not true. The existence of uncountably many uniform countablyinﬁnite Steiner triple systems is known [4], but none of these systems is perfect.Note that an inﬁnite perfect STS cannot be larger than countable since a Hamil-tonian cycle cannot be larger than countable.2 ResultsMany countably inﬁnite structures can be constructed “freely” with any special prop-erties “built in” during the construction. Examples include Steiner systems with largevalues of t, and highly transitive Steiner systems [2]. Here we use such a constructionto obtain a perfect countably inﬁnite Steiner triple system by starting with a suitablepartial STS and carefully “building in” the perfect property during the construction.The following are the steps required in a free construction of a countably inﬁniteSteiner triple system from any ﬁnite (or countably inﬁnite) partial STS.• Start with a ﬁnite (or countably inﬁnite) partial Steiner triple system• Adjoin alternatively:– new blocks incident with those pairs of points not already in a blockINFINITE STEINER TRIPLE SYSTEMS 275– new points so each existing block has 3 points• After countably many steps the result is a Steiner triple system.To obtain a perfect countably inﬁnite Steiner triple system we must ensure thatno cycle graph G(a, b) in our system contains a ﬁnite cycle, and that each cycle graphG(a, b) is connected. Thus we start with a partial STS with no cycle graph G(a, b)containing a ﬁnite cycle. At each stage of the construction we insist that any newblock contains at least one new point. This rules out any ﬁnite cycle being created inany cycle graph G(a, b) because if the block axy creates a cycle in G(a, b) then thereexists an xy-path in G(a, b) say [x, w, v, ..., z, y] where possibly v = z. But then wehave blocks avw, bwx, byz; so all of a, x, y are existing points, and axy cannot be anew block. In this way, all that we need to check is that the result is a countablyinﬁnite Steiner triple system and that all the cycle graphs G(a, b) are connected.Theorem 2.1 There exists a countably infinite perfect Steiner triple system.Proof We start with a (ﬁnite or countably inﬁnite) partial STS with no cycle graphG(a, b) containing a ﬁnite cycle and we construct a countably inﬁnite perfect Steinertriple system in stages.At even-numbered stages, we list all pairs of existing points, and for each suchpair ab which is not already contained in a block, we add a new point n and a blockabn.At odd-numbered stages, we list all pairs of existing points. For each such pair ab,consider the cycle graph G(a, b). This will be a union of paths and isolated vertices.Suppose that x and y are either ends of paths or isolated vertices (if both are pathends, it should not be the same path). Now add new vertices and blocks as follows:• If x is isolated or on an a-coloured edge and y is isolated or on a b-colourededge, add blocks bxn and any, where n is a new point. Similarly if the coloursare reversed.• If x and y are on a-coloured edges, then add blocks bxn1, an1n2, and bn2y.Similarly if they are both on b-coloured edges.An easy check shows that we never introduce a second block containing two pointsalready in a block. (In the ﬁrst case, for example, if there were a block containing band x, then x would not be isolated or the end of a path in G(a, b).)Now given any two points, there will be a ﬁrst stage of the construction at whichthey have both appeared; and in the next even-numbered stage, a block containingthem will be introduced. So we construct a Steiner triple system S.Given points a and b, and x and y in V \ {a, b, c}, where abc is a block, at thenext odd-numbered stage after all these points have been introduced, x and y willbe in the same connected component of the cycle graph G(a, b). So the countablyinﬁnite Steiner triple system produced is perfect.To construct non-isomophic countably inﬁnite perfect Steiner triple systems inthis way we need to choose the partial STS used at the start of the construction276 P.J. CAMERON AND B.S. WEBBcarefully. The inﬁnite family of partial STS used to construct the uniform count-ably inﬁnite Steiner triple systems in [4] are ‘perfect’ for our needs. We take theconstruction from the proof of Lemma 2.3 in that paper.Let D1 be a partial STS with P (D1) = {a, b, c, d, e, f, g, h, i} and the six blocks:{a, b, c}, {d, e, f}, {g, h, i}, {a, d, g}, {b, e, h}, {c, f, i}.It is connected and has 9 points all of degree 2.We form an inﬁnite sequence of partial STS called D1,D2,D3 . . . starting fromD1. We construct Di+1 from Di in the following manner. Di is a connected partialSTS with 3(i + 2) points all of degree 2, where i ≥ 1. There exist two points ofP (Di), say x, y, that do not lie together in a block of Di. Replace x by a new pointx1 in one of the two blocks containing x, and replace x by another new point x2 inthe other block containing x. Carry out a similar replacement of y by new points y1and y2. This may disconnect the conﬁguration, but we can assume that x1 remainsin the same component as y1. Now introduce a new point z and two new blocksx1y2z and x2y1z. The resulting partial STS Di+1 has 3(i+ 3) points, all of degree 2,and is connected. Clearly no Di can contain any other Dj as a (partial) sub-system.We consider the cycle graphs G(a, b) of Di; because the degree of each pointis 2, there are at most 4 edges in each such graph (when the points a and b arenot contained together in a block). The shortest cycle possible in any cycle graphG(a, b) is of length 4 and occurs only when the STS or partial STS contains a Paschconﬁguration (also called a quadrilateral) with four blocks of the form{a, x, y}, {b, y, z}, {a, z, t}, {b, x, t}.The way that the partial STS Di are constructed precludes this possibility, so noneof the cycle graphs G(a, b) of any of the Di contains a cycle.Let D0 be the trivial partial STS comprising just a single point and no blocks.Denote by C the partial STS formed from some collection of partial STS Di (i ≥ 0);note that they need not all be distinct.Theorem 2.2 There exists a countably infinite perfect Steiner triple system contain-ing the partial STS C, which contains no other connected finite partial sub-system allof whose points have degree 2.Proof The cycle graphs G(a, b) of the partial STS C contain no cycles since eachcomponent of this partial STS is simply one of the Di and there are no edges in anyG(a, b) between these components.We therefore follow the construction in stages, as in the proof of Theorem 2.1,starting with the partial STS C. The resulting countably inﬁnite Steiner triple systemwill be perfect by Theorem 2.1.At every stage in the construction, every new block added contains a new point,and so no new ﬁnite partial sub-system can be formed all of whose points are ofdegree 2, since there must be at least one new point of degree 1 on every blockadded.INFINITE STEINER TRIPLE SYSTEMS 277Corollary 2.2.1 There are 2ℵ0 non-isomorphic countably infinite perfect Steinertriple systems.Proof If two distinct inﬁnite subsets of {D1,D2,D3, . . .} are taken to form twoinitial partial STS used in two applications of the construction, then the two result-ing perfect countably inﬁnite STS are non-isomorphic since they contain diﬀerentnumbers of the partial STS Di. There are 2ℵ0 such subsets, so there are that manynon-isomorphic countably inﬁnite STS.3 Concluding RemarksThe existence of these perfect countably inﬁnite Steiner triple systems ﬁts with theassertion that the question of the existence of countably inﬁnite Steiner systems is ingeneral inordinately simpler than the ﬁnite case, since there is usually “space” to ﬁteverything in. Ad hoc methods, “free” constructions and more general constructionsusing Model Theoretic results, such as the Fra¨ısse´ Limit can be used to constructmany countably inﬁnite Steiner systems with speciﬁed properties.Care has to be exercised though, as some inﬁnite systems with speciﬁed propertiesdo not exist. On one hand, the existence of some systems is easily ruled out bycombinatorial restrictions; for example, the non-existence of a block-transitive, point-intransitive Steiner triple system [1, 9], although block-transitive, point-intransitiveSteiner systems with block size greater than 3 do exist [6]. On the other hand, somecountably inﬁnite Steiner systems with speciﬁed properties do not exist and there isno obvious combinatorial reason why. It has recently been shown that all inﬁniteSteiner systems with block size strictly less than v are resolvable [5], and so no non-resolvable inﬁnite Steiner triple system exists. Note, however, that the projectiveplane is an example of a non-resolvable Steiner system with block size equal to v.AcknowledgmentsThis work was undertaken during the 35th ACCMCC held at Monash University,where both authors were invited speakers.References[1] P. J. Cameron, An orbit theorem for Steiner triple systems, Discrete Math. 125(1994), 97–100.[2] P. J. Cameron, Infinite versions of some topics in finite geometry, “Geometricalcombinatorics,” Research Notes in Mathematics Series 114 (Eds.) F. C. Holroydand R. J. Wilson, Pitman, Boston, 1984, pp. 13–20.[3] P. J. Cameron (ed.), Research problems from the BCC22, Discrete Math. 311(2011), 1074–1083. Problem 506 (BCC22.2) proposed by M. J. Grannell, T. S.Griggs and B. S. Webb.278 P.J. CAMERON AND B.S. WEBB[4] K. M. Chicot, M. J. Grannell, T. S. Griggs and B. S. Webb, On sparse countablyinﬁnite Steiner triple systems, J. Combin. Des. 18 (2010), 115–122.[5] P. Danziger, D. Horsley and B. S. Webb, Resolvability of Inﬁnite Designspreprint.[6] D. M. Evans, Block transitive Steiner systems with more than one block orbit,J. Combin. Des. 12 (2004), 459–465.[7] A. D. Forbes, M. J. Grannell and T. S. Griggs, On 6-sparse Steiner triple sys-tems, J. Combin. Theory Ser. A 114 (2007), 235–252.[8] M. J. Grannell, T. S. Griggs and J. P. Murphy, Some new perfect Steiner triplesystems, J. Combin. Des. 7 (1999), 327–330.[9] B. S. Webb, Orbits of inﬁnite block designs, Discrete Math. 169 (1997), 287–292.(Received 4 Feb 2012)

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Perfect countably infinite Steiner triple systems

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