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# On the existence of $d$-homogeneous $3$-way Steiner trades

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## Abstract

A $\mu$-way $(v, k, t)$ trade $T = \{T_{1} , T_{2}, . . ., T_{\mu} \}$ of volume $m$ consists of $\mu$ disjoint collections $T_{1}, T_{2}, \ldots, T_{\mu}$, each of $m$ blocks 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 $T_{i}$ (for $1 \leq i \leq \mu$). A $\mu$-way $(v, k, t)$ trade is called $\mu$-way $(v, k, t)$ Steiner trade if any $t$-subset of found$(T)$ occurs at most once in $T_{1}$ $(T_{j},\ j \geq 2)$. A $\mu$-way $(v,k,t)$ trade is called $d$-homogeneous if each element of $V$ occurs in precisely $d$ blocks of $T_{1}$ $(T_{j},~ j \geq 2)$. 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 $d\in \{4,5,6\}$, except for seven small values of $v$.
On the existence of d-homogeneous 3-way Steiner trades
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
1iµ). 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.
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 iµ). In
other words any pair of collections {Ti, Tj}, 1 i<jµis a (v, k, t) trade of volume
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 xfound(T), the number of blocks in Ti(for
1iµ) which contains xis denoted by rx. The set of blocks in Ti(for 1 iµ) which
contains xis denoted by Tix (for 1 iµ). 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 (v1, 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
1, T
2, ..., T
µ}is called a subtrade of T={T1, T2, ..., Tµ}, if T
iTi
for 1 iµ.
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)
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 v0 (mod 3), so we use
some obtained results on 3-way d-homogeneous Latin trades.
ALatin square of order nis an n×narray L= (ij) 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
which is obtained from another one by deleting its empty rows and empty columns, is
called a µ-way d-homogeneous Latin trade (for µd) or brieﬂy a (µ, d, m) Latin trade,
if it has mrows and in each row and each column Lrfor 1 rµ, 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 4dm)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) 6d13,
(d) d= 15,
(e) d4and md2,
(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 v2d+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)
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={T1T
1, T2T
2, T3T
3}is a 3-way d-homogeneous (v1+v2,3,2) Steiner
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
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 4dm)exist
for
3
(a) d= 4, except possibly for m= 6,7and 11,
(b) d= 5, except possibly for m= 6,
(c) 6d13,
(d) d= 15,
(e) d4and md2,
(f) m0 (mod 5), except possibly for m= 30,
(g) m0 (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 v1,3 (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 ((v1)/2)-homogeneous (v, 3,2) Steiner trade of
volume v(v1)/6for every v1,3 (mod 6),v̸= 7.
Proof. In [5], it is shown that there exist 3 disjoint ST S (v) for every v1,3 (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= 9, where ∈ {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= 8, where ∈ {1,2,3,· · · }, there exists a 3-way 3-homogeneous
(v, 3,2) Steiner trade of volume v.
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= 9+ 8, 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
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 xfound(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
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-
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
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
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
v8), 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] = {cZ|acb}.
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 Tof volume 8 or 9, respectively. Therefore, T\Tis 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 Tof volume 8 or 9, respectively. So T\Tis 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 v8, 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 v8 in the form 9+8,
where ℓ, ℓ0 as follows:
v= 9k, where k1
v= 9k+ 1 = 9(k7) + 64, where k70
v= 9k+ 2 = 9(k6) + 56, where k60
v= 9k+ 3 = 9(k5) + 48, where k50
v= 9k+ 4 = 9(k4) + 40, where k40
v= 9k+ 5 = 9(k3) + 32, where k30
v= 9k+ 6 = 9(k2) + 24, where k20
v= 9k+ 7 = 9(k1) + 16, where k10
v= 9k+ 8, where k0
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
v9and v0 (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 v12 and v0 (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, B1B2) 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 v13, except possibly for v∈ {17,19,20,22,23,25}.
Proof. By Remark 2, v13. We investigate three cases for v. For v= 3, where
6, the result clearly follows by Theorem 1. For cases v= 3+ 1 = 3(4) + 13 and
v= 3+ 2 = 3(4) + 14, where 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, m3, (by Theorem 7)
(b) d= 5, m4 except possibly for m= 6, (by Theorem 8)
(c) 7d13, md, (by Theorem 1)
(d) d= 15, md, (by Theorem 1)
(e) d4 and md2, (by Theorem 1)
(f) m0 (mod 5) and md, except possibly for m= 30, (by Theorem 1)
(g) m0 (mod 7) and md, except possibly for m= 7 (where d= 4) and
m= 42. (by Theorem 1)
10
(II) v= 6m+ 1:
d= 3m,m2. (by Theorem 2)
(III) v= 6m+ 3:
d= 3m+ 1, m1. (by Theorem 2)
(IV) All v:
(a) d= 3, v8 except for v[10,15] [19,23] [28,31] [37,39] ∪ {46,47,55},
(by Theorem 6)
(b) d= 6, v13 except possibly for v∈ {17,19,20,22,23,25}. (by Theorem 9)
References
[1] P. Adams, E. J. Billington, D. E. Bryant, and E. S. Mahmoodian. On the possible
volumes of µ-way Latin trades. aequationes mathematicae, 63(3):303–320, 2002.
[2] B. Bagheri Gh., D. M. Donovan, and E. S. Mahmoodian. On the existence of 3-way
k-homogeneous Latin trades. Discrete Mathematics, 312(24):3473–3481, 2012.
[3] H. Cao, L. Ji, and L. Zhu. Large sets of disjoint packings on 6k+5 points. Journal of
Combinatorial Theory, Series A, 108(2): 169–183, 2004.
[4] N. J. Cavenagh, D. M. Donovan and E. S¸. Yazici. Minimal homogeneous Steiner
2(v, 3) trades. Discrete Mathematics, 308(5-6):741–752, 2008.
[5] S. Milici and G. Quattrocchi. On the intersection problem for three Steiner triple
systems. Ars Combinatoria, 24A:175–194, 1987.
[6] S. Rashidi and N. Soltankhah. On the possible volume of three way trades. Electronic
Notes in Discrete Mathematics, 43:5–13, 2013.
[7] D. R. Stinson. A survey of Kirkman triple systems and related designs. Discrete
Mathematics, 92(1-3):371–393, 1991.
11
... Not much is known for the mentioned questions about μ-way (v, k, t) trades for μ ≥ 3 and most of the papers have been focused on the case μ = 2. Some questions have been answered about the existence and non-existence of 3-way (v, k, t) trades for some special values of k and t , see [1,5,10]. Let S 3s (t, k) denote the set of all possible volume sizes of a 3-way (v, k, t) Steiner trade. ...
... The collection D is a RB (9,3,1). The blocks are written as columns. ...
... Proof Some volumes are multiples of three or four with the conditions of Theorem 1.5. Other volumes can be written as m1 + m 2 from previous parts. By this method, there exists a 3-way (v, 13, 2) Steiner trade of volumes {88, 94, 118, 119, 122, 125, 127, 131, 133, 134, 137, 139, 142, 142, 143, 145, 146, 149, 151, 154, 155, 157, 158, 161, 163, 166, 167, 170, 171, 173, 175} and m = 36 + m , for m ≥ 144. ...
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A 3-way (v,k,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v,k,t)$\end{document} trade T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T$\end{document} of volume m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m$\end{document} consists of three pairwise disjoint collections T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{1}$\end{document}, T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{2}$\end{document} and T3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{3}$\end{document}, each of m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m$\end{document} blocks of size k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k$\end{document}, such that for every t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t$\end{document}-subset of v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v$\end{document}-set V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V$\end{document}, the number of blocks containing this t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t$\end{document}-subset is the same in each Ti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{i}$\end{document} for 1≤i≤3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq i\leq 3$\end{document}. If any t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t$\end{document}-subset of found(T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T$\end{document}) occurs at most once in each Ti\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{i}$\end{document} for 1≤i≤3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq i\leq 3$\end{document}, then T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T$\end{document} is called 3-way (v,k,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v,k,t)$\end{document} Steiner trade. We attempt to complete the spectrum S3s(v,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{3s}(v,k)$\end{document}, the set of all possible volume sizes, for 3-way (v,k,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v,k,2)$\end{document} Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and r×s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r\times s$\end{document} packing grid blocks. Previously, we obtained some results about the existence some 3-way (v,k,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v,k,2)$\end{document} Steiner trades. In particular, we proved that there exists a 3-way (v,k,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v,k,2)$\end{document} Steiner trade of volume m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m$\end{document} when 12(k−1)≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$12(k-1)\leq m$\end{document} for 15≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$15\leq k$\end{document} (Rashidi and Soltankhah in Discrete Math. 339(12): 2955–2963, 2016). Now, we show that the claim is correct also for k≤14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\leq 14$\end{document}.
... The minimum volume and minimum foundation size of (v, k, t) trade was obtained by Hwang [11] and recently the minimum volume and minimum foundation size of µ-way (v, 3, 2) trades for each integer number µ ≥ 3 has been studied in [7]. Some results on d-homogeneous µ-way (v, 3, 2) Steiner trades for µ = 2, 3 can be found in [1,3]. In this paper, we will focus on d-homogeneous µ-way (v, 3, 2) Steiner trades for each d ≡ 0 (mod 3). ...
... One can easily seen that v ∈ A q . The second part was proved in [1]. Then the assertion is true. ...
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... Not much is known for the mentioned questions about µ-way (v, k, t) trades for µ ≥ 3 and most of the papers have been focused on the case µ = 2. Some questions have been answered about the existence and non-existence of 3-way (v, k, t) trades for some special values of k and t, see [1,5,10]. Let S 3s (t, k) denote the set of all possible volume sizes of a 3-way (v, k, t) Steiner trade. ...
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A 3-way $(v,k,t)$ trade $T$ of volume $m$ consists of three pairwise disjoint collections $T_1$, $T_2$ and $T_3$, each of $m$ blocks 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 $T_i$ for $1\leq i\leq 3$. If any $t$-subset of found($T$) occurs at most once in each $T_i$ for $1\leq i\leq 3$, then $T$ is called 3-way $(v,k,t)$ Steiner trade. We attempt to complete the spectrum $S_{3s}(v,k)$, the set of all possible volume sizes, for 3-way $(v,k,2)$ Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and $r\times s$ packing grid blocks. Previously, we obtained some results about the existence some 3-way $(v,k,2)$ Steiner trades. In particular, we proved that there exists a 3-way $(v,k,2)$ Steiner trade of volume $m$ when $12(k-1)\leq m$ for $15\leq k$ (Rashidi and Soltankhah, 2016). Now, we show that the claim is correct also for $k\leq 14$.
... Some questions have been answered about the existence and non-existence of 3-way (v, k, t) trades for some special values of k and t in [22]. Also, some results on the 3-way d-homogeneous (v, 3, 2) Steiner trades for some special values of d have been obtained in [1]. ...
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A 3-way (v, k, t) trade of volume s consists of 3 disjoint collections and , each of s blocks, such that for every t-subset of v-set V, the number of blocks containing this t-subset is the same in each ( ). In this paper we prove the existence of: (i) 3-way (v, k, 1) trades (Steiner trades) of each volume . (ii) 3-way (v, k, 2) trades of each volume except . We establish the non-existence of 3-way (v, 3, 2) trade of volume 7. It is shown that the volume of a 3-way (v, k, 2) Steiner trade is at least 2k for . Also the spectrum of 3-way ( ) Steiner trades for and 4 are specified.
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A (2,3)-packing on X is a pair , where is a set of 3-subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X,E) such that E consists of all the pairs which do not appear in any block of . For a (6k+5)-set X a large set of maximum packing, denoted by LMP(6k+5), is a set of 6k+1 disjoint (2,3)-packings on X with a cycle of length four as their common leave. Schellenberg and Stinson (J. Combin. Math. Combin. Comput. 5 (1989) 143) first introduced such a large set problem and used it to construct perfect threshold schemes. In this paper, we show that an LMP(6k+5) exists for any positive integer k. This complete solution is based on the known existence result of S(3,4,v)s by Hanani and that of 1-fan S(3,4,v)s and S(3,{4,5,6},v)s by the second author. Partitionable candelabra system also plays an important role together with two special known LMP(6k+5)s for k=1,2.
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A Steiner 2-(v,3) trade is a pair (T1,T2) of disjoint partial Steiner triple systems, each on the same set of v points, such that each pair of points occurs in T1 if and only if it occurs in T2. A Steiner 2-(v,3) trade is called d-homogeneous if each point occurs in exactly d blocks of T1 (or T2). In this paper we construct minimal d-homogeneous Steiner 2-(v,3) trades of foundation v and volume dv/3 for sufficiently large values of v. (Specifically, v>3(1.75d2+3) if v is divisible by 3 and v>d(4d/3+1+1) otherwise.)
The result is followed by Theorem 1, Proposition 2 and Remarks 1 and 2
• Proof
Proof. The result is followed by Theorem 1, Proposition 2 and Remarks 1 and 2.
According to Theorem 2, there exists a 3-way 4-homogeneous (9, 3, 2) Steiner trade of volume
• Proof
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