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On the existence of d-homogeneous 3-way Steiner trades
H. Amjadi and N. Soltankhah ∗
Faculty of Mathematical Sciences
Alzahra University
Vanak Square 19834, Tehran, Iran
Abstract
Aµ-way (v, k, t) trade T={T1, T2, ..., Tµ}of volume mconsists of µdisjoint
collections T1, T2, ..., Tµ, each of mblocks of size k, such that for every t-subset of
v-set Vthe number of blocks containing this t-subset is the same in each Ti(for
1≤i≤µ). A µ-way (v, k, t) trade is called µ-way (v, k, t) Steiner trade if any t-subset
of found(T) occurs at most once in T1(Tj, j ≥2). A µ-way (v, k, t) trade is called
d-homogeneous if each element of Voccurs in precisely dblocks of T1(Tj, j ≥2). In
this paper we characterize the 3-way 3-homogeneous (v, 3,2) Steiner trades of volume
v. Also we show how to construct a 3-way d-homogeneous (v, 3,2) Steiner trade for
d∈ {4,5,6}, except for seven small values of v.
Key words: Steiner trade, µ-way trade, Homogeneous trade.
Subject classification: 05B05.
1 Introduction and preliminary results
Let Vbe a set of velements and kand tbe two positive integers such that t<k<v. A
(v, k, t)trade T={T1, T2}of volume mconsists of two disjoint collections T1and T2, each
of containing m,k-subsets of V, called blocks, such that every t-subset of Vis contained
in the same number of blocks in T1and T2. A (v, k, t) trade is called (v, k, t)Steiner trade
if any t-subset of Voccurs in at most once in T1(T2). In a (v, k, t) trade, both collections
of blocks must cover the same set of elements. This set of elements is called the foundation
of the trade and is denoted by found(T).
The concept of µ-way (v, k, t) trade, was defined recently in [6].
Definition: A µ-way (v, k, t)trade T={T1, T2, ..., Tµ}of volume mconsists of µdisjoint
collections T1, T2, ..., Tµ, each of mblocks of size k, such that for every t-subset of v-set V
the number of blocks containing this t-subset is the same in each Ti(for 1 ≤i≤µ). In
other words any pair of collections {Ti, Tj}, 1 ≤i<j≤µis a (v, k, t) trade of volume
Email addresses: h.amjadi@alzahra.ac.ir (H. Amjadi), soltan@alzahra.ac.ir (N. Soltankhah)
∗Corresponding author
1
PLEASE CITE AS: H. Amjadi and N. Soltankhah. On the existence d-homogeneous 3-way
Steiner trades, Utititas Mathematica, 108 (2016), 307-320.
m. It is clear by the definition that a trade is a 2-way trade. A µ-way (v, k, t) trade is
called µ-way (v, k, t)Steiner trade if any t-subset of found(T) occurs at most once in T1
(Tj, j ≥2). A µ-way (v, k, t) trade is called d-homogeneous if each element of Voccurs
in precisely dblocks of T1(Tj, j ≥2). Let x∈found(T), the number of blocks in Ti(for
1≤i≤µ) which contains xis denoted by rx. The set of blocks in Ti(for 1 ≤i≤µ) which
contains xis denoted by Tix (for 1 ≤i≤µ). It is easy to see that Tx={T1x, ..., Tµx}
is a µ-way (v, k, t −1) trade of volume rx. If we remove xfrom the blocks of Tx, then
the result will be a µ-way (v−1, k −1, t −1) trade which is called the derived trade of T.
Also it is easy to show that if Tis a Steiner trade then its derived trade is also a Steiner
trade. A trade T′={T′
1, T ′
2, ..., T ′
µ}is called a subtrade of T={T1, T2, ..., Tµ}, if T′
i⊆Ti
for 1 ≤i≤µ.
For µ= 2, Cavenagh et al. [4], constructed minimal d-homogeneous (v, 3,2) Steiner
trades of foundation vand volume dv/3 for sufficiently large values of v, (specifically,
v > 3(1.75d2+ 3) if vis divisible by 3 and v > d(4d/3+1 + 1) otherwise).
Generally we can ask the following question.
Question 1. For given µ,dand v, does there exist a µ-way d-homogeneous (v, 3,2)
Steiner trade?
In this paper, we aim to construct 3-way d-homogeneous (v, 3,2) Steiner trades. The
Latin trades are a useful tool for building these trades when v≡0 (mod 3), so we use
some obtained results on 3-way d-homogeneous Latin trades.
ALatin square of order nis an n×narray L= (ℓij) usually on the set N={1,2, ..., n}
where each element of Nappears exactly once in each row and exactly once in each column.
We can represent each Latin square as a subset of N×N×N,L={(i, j;k)|element k
is located in position (i, j )}. A partial Latin square of order nis an n×narray P= (pij)
of elements from the set Nwhere each element of Nappears at most once in each row
and at most once in each column. The set SP={(i, j )|(i, j;k)∈P}of the partial Latin
square Pis called the shape of Pand |SP|is called the volume of P.
Aµ-way Latin trade, (L1, L2, ..., Lµ), of volume sis a collection of µpartial Latin
squares L1, L2, ..., Lµcontaining exactly the same sfilled cells, such that if cell (i, j) is
filled, it contains a different entry in each of the µpartial Latin squares, and such that
row iin each of the µpartial Latin squares contains, set-wise, the same symbols and
column j, likewise. Adams et al.[1] studied µ-way Latin trades. A µ-way Latin trade
which is obtained from another one by deleting its empty rows and empty columns, is
called a µ-way d-homogeneous Latin trade (for µ≤d) or briefly a (µ, d, m) Latin trade,
if it has mrows and in each row and each column Lrfor 1 ≤r≤µ, contains exactly d
elements, and each element appears in Lrexactly dtimes. Bagheri et al.[2], studied the
µ-way d-homogeneous Latin trades and their main result is as follows:
Theorem A.[2] All (3, d, m)Latin trades (for 4≤d≤m)exist, for
(a) d= 4, except for m= 6 and 7and possibly for m= 11,
(b) d= 5, except possibly for m= 6,
2
(c) 6≤d≤13,
(d) d= 15,
(e) d≥4and m≥d2,
(f) ma multiple of 5, except possibly for m= 30,
(g) ma multiple of 7, except possibly for m= 42 and (3,4,7) Latin trade.
All 3-way (v, 3,2) Steiner trades are characterized in [6]. The authors proved that there
is no 3-way (v, 3,2) Steiner trade of volumes 1,2,3,4,5 and 7. Also they showed that the
3-way (v, 3,2) Steiner trade of volume 6 is unique (where the number of occurrences of
each element is not the same). So the following proposition is clear.
Proposition 1. The 3-way d-homogeneous (v, 3,2) Steiner trade of volume mdoes not
exist for m∈ {1,2, ..., 7}.
Remark 1. Since the volume of a µ-way d-homogeneous (v, 3,2) Steiner trade is dv/3,
at least one of dor vshould be multiple of 3.
Remark 2. In a µ-way d-homogeneous (v, 3,2) Steiner trade, since every element should
belong to dblocks and the other elements of these blocks should be different, so v≥2d+1.
Lemma 1. If there exist two 3-way d-homogeneous (v1,3,2) and (v2,3,2) Steiner trades
of volume m1and m2, respectively, then we have a 3-way d-homogeneous (v1+v2,3,2)
Steiner trade of volume m1+m2.
Proof. Let T={T1, T2, T3}be a 3-way d-homogeneous (v1,3,2) Steiner trade of volume
m1and T′={T′
1, T ′
2, T ′
3}be a 3-way d-homogeneous (v2,3,2) Steiner trade of volume m2.
It is enough to relabel the elements of found(T′), such that found(T)∩found(T′) = ∅. It
is clear that S={T1∪T′
1, T2∪T′
2, T3∪T′
3}is a 3-way d-homogeneous (v1+v2,3,2) Steiner
trade of volume m1+m2.□
The following lemma which is similar to Lemma 2 of [4], shows how to construct a
3-way d-homogeneous (3m, 3,2) Steiner trade, by using 3-way d-homogeneous Latin trade
of order m.
Lemma 2. Let (L1, L2, L3)be a 3-way d-homogeneous Latin trade of order m. For each
α∈ {1,2,3}, define Tα={{i1, j2, k3}| (i, j;k)∈Lα}. Then T={T1, T2, T3}is a 3-way
d-homogeneous (3m, 3,2) Steiner trade.
The following theorem is a consequence of Theorem A and Lemma 2.
Theorem 1. All 3-way d-homogeneous (3m, 3,2) Steiner trades (for 4≤d≤m)exist
for
3
(a) d= 4, except possibly for m= 6,7and 11,
(b) d= 5, except possibly for m= 6,
(c) 6≤d≤13,
(d) d= 15,
(e) d≥4and m≥d2,
(f) m≡0 (mod 5), except possibly for m= 30,
(g) m≡0 (mod 7), except possibly for m= 7 (where d= 4) and m= 42.
The existence of 3-way d-homogeneous (v, 3,2) Steiner trades when vis not multiple
of 3, will be investigated later.
ASteiner triple system of order v(briefly ST S (v)) is a pair (X, B) where Xis a v-
set and Bis a collection of 3-subsets of X(called triple) such that every pair of distinct
elements of Xbelongs to exactly one triple of B. It is well known that a ST S (v) exists
if and only if v≡1,3 (mod 6). A Kirkman triple system of order v(briefly K T S(v)) is a
Steiner triple system of order v, (X, B) together with a partition Rof the set of triples B
into subsets R1, R2, ..., Rncalled parallel classes such that each Ri(for i= 1,2, ..., n) is a
partition of X.
Theorem 2. There exists a 3-way ((v−1)/2)-homogeneous (v, 3,2) Steiner trade of
volume v(v−1)/6for every v≡1,3 (mod 6),v̸= 7.
Proof. In [5], it is shown that there exist 3 disjoint ST S (v) for every v≡1,3 (mod 6),
v̸= 7. It is obvious that the 3-way trade T={T1, T2, T3}, where T1, T2and T3are the
disjoint ST S (v)s, is the desired trade. □
23-way 3-homogeneous (v, 3,2) Steiner trades
In this section we answer Question 1 when µ=d= 3. Note that by Remark 1 there is no
restriction for vin this case.
Lemma 3. For every v= 9ℓ, where ℓ∈ {1,2,3,· · · }, there exists a 3-way 3-homogeneous
(v, 3,2) Steiner trade of volume v.
Proof. According to Corollary 1 of [2], there exists a (3,3, m) Latin trade if and only if
3|m. By Lemma 2 we can obtain a 3-way 3-homogeneous (v, 3,2) Steiner trade of volume
vfrom a (3,3, m) Latin trade, where v= 3m.□
Lemma 4. For every v= 8ℓ, where ℓ∈ {1,2,3,· · · }, there exists a 3-way 3-homogeneous
(v, 3,2) Steiner trade of volume v.
Proof. The following trade is a 3-way 3-homogeneous (8,3,2) Steiner trade.
4
T1123 147 158 248 267 357 368 456
T2124 138 157 237 268 467 458 356
T3127 135 148 246 238 367 457 568
So the proof is obvious by Lemma 1. □
The last two lemmas and Lemma 1 yields the following theorem.
Theorem 3. For every nonzero v= 9ℓ+ 8ℓ′, where ℓ, ℓ′∈ {0,1,2,3,· · · }, there exists a
3-way 3-homogeneous (v, 3,2) Steiner trade of volume v.
The following lemma can be used for characterizing 3-way 3-homogeneous (v, 3,2) Steiner
trades of volume v.
Lemma 5. There exist only two non-isomorphic 3-way (v, 2,1) Steiner trade of volume
3.
Proof. Let T= (T1, T2, T3) be a 3-way (v, 2,1) Steiner trade of volume 3. Since Tis a
Steiner trade, |found(T)|= 6 and let found(T) = {1,2, ..., 6}. Without loss of generality
we can assume that T1={12,34,56}and T2={13,26,45}. Now if we consider the blocks
of the trade as edges in as 6 vertex graph and color the edges of each trade with a different
color, then the first two trades form an alternating colored 6-cycle. Up to isomorphism,
there are only two different one factors in K6to complete this graph to a 3-regular graph.
So all trademates put together either form a bipartite or a non-bipartite 3-regular graph.
So T= (T1, T2, T3) up to isomorphism, should be as one of the following trades:
T1T2T3
12 13 14
34 26 25
56 45 36
T1T2T3
12 13 15
34 26 24
56 45 36
□
Theorem 4. Every 3-way 3-homogeneous (v, 3,2) Steiner trade of volume vcontains a
3-way 3-homogeneous (u, 3,2) Steiner trade of volume 8or 9, as a subtrade.
Proof. Let T={T1, T2, T3}be a 3-way 3-homogeneous (v, 3,2) Steiner trade of vol-
ume vwith found(T) = {1,2, ..., v}. Let x∈found(T), then Tx={T1x, T2x, T3x}
is a (v, 3,1) trade of volume rx= 3. Without loss of generality we can assume that
found(Tx) = {x, 1,2, ..., 6}. According to Lemma 5 there exist only two cases for Tx.
T1xT2xT3x
x12 x13 x14
x34 x26 x25
x56 x45 x36
T1xT2xT3x
x12 x13 x15
x34 x26 x24
x56 x45 x36
5
Let Tcontains Txwhich is as the first form.
Since T={T1, T2, T3}is a 3-homogeneous trade, each of T1,T2and T3should contain two
other blocks containing element 1. According to definition of Steiner trades, T1cannot
contain block 134 (since it has block x34). So there exist three possible cases:
1. Only one of T2and T3contains block 123 or 124:
Let T2contains block 123 (If T3contains block 124, then the same result will be
achieved).
T1x12 x34 x56 13a14b
T2x14 x25 x36 123 1ab
T3x13 x26 x45 12b14a
There are four possible cases for elements aand b:
1.1. a, b /∈ {5,6}
The other blocks which contain 4 should be as follows. The pair 3bexists in T3
and does not exist in T2, it is a contradiction with definition of trade.
T1x12 x34 x56 13a14b45a
T2x14 x25 x36 123 1ab 45b4a3
T3x13 x26 x45 12b14a43b
1.2. a= 6 and b= 5
The other blocks of T1and T2which contain 6 should be as follows. It is a
contradiction with definition of trade.
T1x12 x34 x56 136 145 624
T2x14 x25 x36 123 156 624
T3x13 x26 x45 125 146
1.3. a /∈ {5,6}and b= 5
A 3-way 3-homogeneous (8,3,2) is achieved.
T1x12 x34 x56 13a145 52a236 46a
T2x14 x25 x36 123 15a546 26a34a
T3x13 x26 x45 125 14a56a23a364
1.4. a= 6 and b /∈ {5,6}
Only one of the remaining blocks of T2can contain 6, but we should have 62,
64, 65 is T2which is impossible.
T1x12 x34 x56 136 14b
T2x14 x25 x36 123 16b
T3x13 x26 x45 12b146
2. T2and T3contain block 123 and 124, respectively:
The other blocks of T2and T3which contain 1 should be as follows. It is a contra-
diction with definition of trade.
6
T1x12 x34 x56 13a14b
T2x14 x25 x36 123 1ab
T3x13 x26 x45 124 1ab
3. T2and T3do not contain the blocks 123 and 124:
T1x12 x34 x56 13a14b
T2x14 x25 x36 13b12a
T3x13 x26 x45 14a12b
There are four possible cases for elements aand b:
3.1. a, b /∈ {5,6}
A 3-way 3-homogeneous (9,3,2) is achieved.
T1x12 x34 x56 13a14b36b45a62a52b
T2x14 x25 x36 13b12a34a45b65a62b
T3x13 x26 x45 14a12b34b3a6 65b52a
3.2. a= 6 and b= 5
The other blocks of T1,T2and T3which contain 6 should be as follows. So the
other blocks of T2and T3which contain 4 should be 423. It is a contradiction
with definition of trade.
T1x12 x34 x56 136 145 624
T2x14 x25 x36 135 126 645 423
T3x13 x26 x45 146 125 635 423
3.3. a /∈ {5,6}and b= 5
The other blocks which contain 5 should be as follows. So the other blocks of
Twhich contain 2 should be as follows. 36 appears two times in T2. It is a
contradiction with definition of Steiner trade.
T1x12 x34 x56 13a145 523 26a
T2x14 x25 x36 135 12a546 236
T3x13 x26 x45 14a125 536 23a
3.4. a= 6 and b /∈ {5,6}
The other blocks which contain 6 should be as follows. There is a block in T2
which contain 45. 4 appears three times in T1but the pair 45 does not appear
in blocks of T1. It is a contradiction with definition of trade.
T1x12 x34 x56 136 14b624
T2x14 x25 x36 13b126 645
T3x13 x26 x45 146 12b635
For the other case, the same result can be obtained by a similar argument. So it can be
deduced that if there exists a 3-way 3-homogeneous (v, 3,2) Steiner trade of volume v(for
v≥8), then it contains a 3-way 3-homogeneous (8,3,2) or (9,3,2) Steiner trade of volume
8 or 9, respectively. □
The following corollary is the direct result of Theorem 4.
7
Corollary 1. If there exists a 3-way 3-homogeneous (v, 3,2) Steiner trade of volume v,
then it can be represented as a union of disjoint 3-way 3-homogeneous (8,3,2) or (9,3,2)
Steiner trades of volume 8or 9, respectively.
Define [a, b] = {c∈Z|a≤c≤b}.
Theorem 5. The 3-way 3-homogeneous (v, 3,2) Steiner trade of volume vdoes not exist
for v∈[1,7] ∪[10,15] ∪[19,23] ∪[28,31] ∪[37,39] ∪ {46,47,55}.
Proof. By Proposition 1, the 3-way 3-homogeneous (v, 3,2) Steiner trade of volume v
does not exist for v∈[1,7]. Let there exist a 3-way 3-homogeneous (v, 3,2) Steiner trade
Tof volume vfor v∈[10,15]. By Theorem 4, it should contain a 3-way 3-homogeneous
(8,3,2) or (9,3,2) Steiner trade T′of volume 8 or 9, respectively. Therefore, T\T′is a
3-way 3-homogeneous (u, 3,2) Steiner trade of volume u, where u∈[1,7], which is im-
possible. By same argument, let there exist a 3-way 3-homogeneous (v, 3,2) Steiner trade
Tof volume vfor v∈[19,23]. By Theorem 4, it should contain a 3-way 3-homogeneous
(8,3,2) or (9,3,2) Steiner trade T′of volume 8 or 9, respectively. So T\T′is a 3-way
3-homogeneous (u, 3,2) Steiner trade of volume u, where u∈[10,15], which is a con-
tradiction. The same way, we can prove non-existence of the other mentioned trades.
□
Theorem 6. For every v≥8, there exists a 3-way 3-homogeneous (v, 3,2) Steiner trade
of volume v, except for v∈[10,15] ∪[19,23] ∪[28,31] ∪[37,39] ∪ {46,47,55}.
Proof. According to Theorem 3, it is enough to represent every v≥8 in the form 9ℓ+8ℓ′,
where ℓ, ℓ′≥0 as follows:
v= 9k, where k≥1
v= 9k+ 1 = 9(k−7) + 64, where k−7≥0
v= 9k+ 2 = 9(k−6) + 56, where k−6≥0
v= 9k+ 3 = 9(k−5) + 48, where k−5≥0
v= 9k+ 4 = 9(k−4) + 40, where k−4≥0
v= 9k+ 5 = 9(k−3) + 32, where k−3≥0
v= 9k+ 6 = 9(k−2) + 24, where k−2≥0
v= 9k+ 7 = 9(k−1) + 16, where k−1≥0
v= 9k+ 8, where k≥0
Using Theorem 5 completes the proof. □
33-way d-homogeneous (v, 3,2) Steiner trade for d∈ {4,5,6}
In this section we answer Question 1 when µ= 3 and d= 4,5 and 6.
3.1 d= 4
In this subsection we completely answer Question 1 for d= 4. Note that since d= 4 by
Remark 1, vshould be a multiple of 3.
8
Proposition 2. There exist 3-way 4-homogeneous (9,3,2),(18,3,2),(21,3,2) and (33,3,2)
Steiner trades of volume 12,24,28 and 44, respectively.
Proof. According to Theorem 2, there exists a 3-way 4-homogeneous (9,3,2) Steiner
trade of volume 12. By Theorem 1, there exist 3-way 4-homogeneous (12,3,2) and (15,3,2)
Steiner trades of volume 16 and 20, respectively. Since 18 = 9 + 9, 21 = 12 + 9 and
33 = 15 + 18, regarding to Lemma 1, the result follows. □
Theorem 7. There exists a 3-way 4-homogeneous (v, 3,2) Steiner trade if and only if
v≥9and v≡0 (mod 3).
Proof. The result is followed by Theorem 1, Proposition 2 and Remarks 1 and 2. □
3.2 d= 5
In this subsection we solve Question 1 for d= 5, except for v= 18. By Remark 1, vshould
be a multiple of 3.
Proposition 3. There exists a 3-way 5-homogeneous (12,3,2) Steiner trade of volume
20.
Proof. It is enough to use three disjoint decomposition of K12 −I(the graph obtained
from K12 by removing the edges of a perfect matching) into triples. In other words, in the
literature of block designs, it is enough to consider three disjoint compatible (2,3)-packings
on 12 points (see [3]). □
Theorem 8. Except possibly for v= 18, there exists a 3-way 5-homogeneous (v, 3,2)
Steiner trade if and only if v≥12 and v≡0 (mod 3).
Proof. The result follows by Theorem 1, Proposition 3 and Remarks 1 and 2. □
3.3 d= 6
In this subsection we solve Question 1 for d= 6, only six values of vare left undecided.
Note that by Remark 1 there is no restriction for vin this case.
Proposition 4. There exist a 3-way 6-homogeneous (v, 3,2) Steiner trade of volume 2v,
where v= 13,14,15 and 16.
Proof. By Theorem 2 there exist a 3-way 6-homogeneous (13,3,2) Steiner trade of vol-
ume 26.
Let X={1,2,3,4,5,6,7,8,9, a, b, c, d, e, f }and let (X, B) be a K T S(15), where
B={123,48c, 5ae, 6bd, 79f, 145,28a, 3df, 69e, 7bc, 167,29b, 3ce, 4af, 58d, 189,2cf , 356,4be,
7ad, 1ab, 2de, 347,59c, 68f, 1cd, 246,39a, 5bf , 78e, 1ef, 257,38b, 49d, 6ac}
9
We define the following permutations on X:
π1= (1 e)(2 4)(3 7)(5 6)(8 b)(a c)(9 d)
π2= (9 7)(3 d)(4 a)(2 c)(6 8)(5 b)(1 e)
The intersection of three sets B, π1(B) and π2(B) is C={79f, 3df, 4af, 2cf , 68f, 5bf, 1ef }.
So T={T1, T2, T3}is a 3-way 6-homogeneous (14,3,2) Steiner trade of volume 28, where
T1=B\C,T2=π1(B)\Cand T3=π3(B)\C.
Let X={1,2,3,4,5,6,7,8,9, a, b, c, d, e, f }and let (X, B1∪B2) be a KT S (15), where
B1={12f, 345,678,9ab, cde}and B2={36f , 15e, 24a, 7bc, 89d, 9cf, 147,2be, 38a, 56d, 7af,
16c, 258,3bd, 49e, 4df, 18b, 269,37e, 5ac, 8ef, 1ad, 23c, 46b, 579,5bf , 139,27d, 48c, 6ae}
We define permutations π= (6 7 8)(9 b a)(cde) on X:
B2,π(B2) and π(π(B2)) are disjoint. So T={B2, π(B2), π(π(B2))}is a 3-way 6-
homogeneous (15,3,2) Steiner trade of volume 30.
Let Y={1,2,3,4,5,6,7,8,9, a, b, c, d, e, f, g}and
D={59e, 6cg, 7af, 8bd, 5bf, 6ad, 7ce, 89g, 19f, 2bg, 3cd, 4ae, 1ag, 2cf, 3be, 49d, 15d, 28e, 36f,
47g, 16e, 27d, 35g, 48f, 17b, 269,38a, 45c, 18c, 25a, 379,46b}
(In fact Dis the block set of a Kirkman frame of type 44with group set
G={1234,5678,9abc, def g}[7]). We define the following permutations on Y:
π3= (1 2)(3 4)(5 6)(7 8)
π4= (1 3)(2 4)(5 7)(6 8)(9 a)(b c)
D,π3(D) and π4(D) are disjoint. So T={D, π3(D), π4(D)}is a 3-way 6-homogeneous
(16,3,2) Steiner trade of volume 32. □
Theorem 9. There exists a 3-way 6-homogeneous (v, 3,2) Steiner trade of volume 2vfor
every v≥13, except possibly for v∈ {17,19,20,22,23,25}.
Proof. By Remark 2, v≥13. We investigate three cases for v. For v= 3ℓ, where
ℓ≥6, the result clearly follows by Theorem 1. For cases v= 3ℓ+ 1 = 3(ℓ−4) + 13 and
v= 3ℓ+ 2 = 3(ℓ−4) + 14, where ℓ≥10, we use Theorem 1 and Proposition 4.
For v∈ {13,14,15,16,26,28,29}, by Proposition 4 and Lemma 1, there exists a 3-way
6-homogeneous (v, 3,2) Steiner trade of volume 2v.□
Our results are summarized below:
Main Theorem. All 3-way d-homogeneous (v, 3,2) Steiner trades exist for
(I) v= 3m:
(a) d= 4, m≥3, (by Theorem 7)
(b) d= 5, m≥4 except possibly for m= 6, (by Theorem 8)
(c) 7≤d≤13, m≥d, (by Theorem 1)
(d) d= 15, m≥d, (by Theorem 1)
(e) d≥4 and m≥d2, (by Theorem 1)
(f) m≡0 (mod 5) and m≥d, except possibly for m= 30, (by Theorem 1)
(g) m≡0 (mod 7) and m≥d, except possibly for m= 7 (where d= 4) and
m= 42. (by Theorem 1)
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(II) v= 6m+ 1:
d= 3m,m≥2. (by Theorem 2)
(III) v= 6m+ 3:
d= 3m+ 1, m≥1. (by Theorem 2)
(IV) All v:
(a) d= 3, v≥8 except for v∈[10,15] ∪[19,23] ∪[28,31] ∪[37,39] ∪ {46,47,55},
(by Theorem 6)
(b) d= 6, v≥13 except possibly for v∈ {17,19,20,22,23,25}. (by Theorem 9)
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systems. Ars Combinatoria, 24A:175–194, 1987.
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