Intersections of Steiner quadruple systems

Discrete Mathematics North-Holland

104 (1992) 227-244

Intersections systems Alan

Hartman

227

of Steiner quadruple

and Zvi Yehudai

IBM Israel, Science and Technology and Scientific Center, Technion City, Haifa 32oo0, Israel Received 25 January 1990 Revised 14 September 1990

Abstract Hartman, A. and Z. Yehudai, ics 104 (1992) 227-244.

Intersections

of Steiner

quadruple

systems,

Discrete

Mathemat-

We complete the determination of the sets of possible intersection sizes for Steiner quadruple systems of all orders u = 2 or 4 (mod 6), v # 14, 26. For z1= 26 we leave only one possible intersection size undecided. For u = 14 there are 20 undecided values.

1. Introduction A Steiner system (S(t, k, v)) is a pair (Q, q) where Q is a set of points of cardinality 2r, and q is a collection of k-element subsets of Q called blocks such that every t-element subset of Q is contained in exactly one block of q. The number of points, u in Q is the order of the system. Two Steiner systems size 12 if they have W, k, v) sd and (Q, q2) are said to have intersection precisely it common blocks, i.e., lql n q2[ = n. Kramer and Mesner [13] initiated the study of intersection problems for Steiner systems, and in 1975 Lindner and Rosa [14] completely determined the spectrum of possible intersection sizes for Steiner triple systems S(2, 3, v). There has been a great deal of work on the analogous problem for Steiner quadruple systems S(3, 4, v) and we discuss the highlights of this work below. Very recently, Colbourn, Lindner, and Hoffman [2] have solved the problem for systems S(2, 4, v) with v 2 40. A Steiner quadruple system of order v, denoted SQS(V), is an S(3, 4, v). It is well known that an SQS(v) has b, = $(y) blocks. Hanani [7] proved that Steiner quadruple systems of order v exist if and only if u ~2 or 4 (mod6). Let J(v) denote the set of possible intersection sizes for Steiner quadruple systems of order v, and let T(n) = (0, 1, . . . , n}. Now let Z(v) = T(b,, - 14) U {b, - 12, b, - 8, b,). Elsevier

Science

Publishers

B.V.

228

A. Hartman,

Gionfriddo

and Lindner

[4] showed

2. Yehudai

that J(V) c Z(V) for all u = 2 or 4 (mod 6).

Clearly

J(2) = (0) and J(4) = (1). Gionfriddo and Lindner showed that Z(8) = Kramer and Mesner [13] showed that .Z(lO) = (0, 2, 6, 14) = Z(8) and (0, 2, 4, 6, 8, 12, 14, 30). After a great deal of intervening work Lo Faro [17]

showed that J(V) = Z(V) for all TV= 4 or 8 (mod 12), with II 2 16. Recently Colbourn and Hartman [l] determined the majority of intersection sizes for u = 2 or 10 (mod 12), with v 2 38. They showed that for u = 10 (mod 12), with u 2 46, J(V) c Z(V)\ T((v - 10)/6), and that J(V) = Z(V) if a certain pair of designs on 22 points Hartman also showed that for u = 2 (mod 12), with v 2 38,

exists.

Colbourn

and

J(V) c Z(V)\ T((v - 2)(v - 14)/6), and that J(V) = Z(v) if a certain pair of designs on 20 points exists. Another recent paper, by Etzion and Hartman [3] has established that 0 E J(V) for all U= 2 or 4 (mod 6), with TV2 8. In this paper we construct the pair of designs on 22 points needed by Colbourn and Hartman to complete the case where v = 10 (mod 12), and Y 3 46. We also apply the methods of Etzion and Hartman to solve the intersection problem for u = 2 (mod 12), with u 3 38. We also determine that 5(22) = Z(22), J(34) = Z(34), and give all but one value in the set Z(26). We have been unable to improve the results of Lo Faro and Puccio [19] and Gionfriddo and Lo Faro [5] who showed that T(47) U (49, 51, 53, 54,55,56,

59,61,91}

cZ(14)

c T(75) u (91).

So the final determination of the sets J(14) and .Z(26) remain outstanding intersection problems for Steiner quadruple systems.

as the

only

2. Preliminaries In this section we introduce some problems and configurations related to the intersection problem for SQS, and show how they are applied to obtain different intersection sizes. We shall use the notation A, + A2 + Let A,, AZ,. . . , A, be sets of integers. . . . +A,, and also C:=, Ai to denote the set of integers {Cy=, ai: Ui EAT}. The notation nA, for an integer 12 and a set of integers A will denote the set of integers {na: a EA}. Note that, with this notation 2A #A +A (unless A is a singleton), for example, 2{1,2} = {2,4} Z {1,2} + {1,2} = {2,3,4}. Let JP(n) be the set of possible numbers of places that two permutations on n letters can agree. It is immediate that JP(n) = T(n)\{n - l}, for all n 2 1.

Intersectionsof Steiner quadruplesystems

A hole system

of order

Y with a hole of order

229

s, denoted

by HQS(V :s),

is a

triple (X, S, q) where X is a set of size V, S is a subset of X of size s, and q is a set of 4-subsets of X, called blocks, such that every 3-subset T c X with (T II SI < 3 is contained in a unique block, and no 3-subset of S is contained in any block. If (X, q) is an SQS(u) (S, T) is an

SQS(s))

with a subsystem then

one

can

construct

deleting the blocks of the subsystem. The intersection size of two hole systems point and hole sets is defined

(S, r) of order s (i.e., an

HQS(u

S c X, r c q, and

: s)

(X, S, q\r)

by

(X, S, ql) and (X, S, q2) with the same

to be the number

of common

blocks,

i.e.,

Let JH(v: s) denote the set of possible intersection sizes of hole systems v with a hole of order s. The number of blocks in a hole system HQS( v : s) is

lql f~ q21.

of order

f[(;)-(;)]=bu-b,. Note that JH(v : 2) =J(v) since a 2-subset S contains no 3-subsets, and that JH(v : 4) = (J(V) \ (0)) + {-l} since a system with a hole of size 4 can be completed to an SQS by adding the hole as an additional block. In somewhat more generality, if we have two hole systems HQS(u :s) which intersect in j blocks, and two Steiner systems SQS(s) which intersect in k blocks then we can construct two SQS(u) which intersect in j + k blocks by the replacement property for subsystems. Thus we have the following.

Lemma 2.1. JH(v : s) + J(s) c J(v).

A candelabra system of order v with a candelabra of type (n, g : s), denoted by CQS(n, g :s), is a quadruple (X, S, y, q) where X is a set of size v = ng + s, S is a subset of X of size s, and y = (G,, G2, . . . , G,) is an ordered partition of X\S into n sets Gi, each of which is of size g. The set q contains 4-subsets of X, called blocks, such that every 3-subset T CX with IT fl (S U Gi)l <3 for all i = in a unique block, and no 3-subset of S U Gi is contained 1, 2, . . . ) n, is contained in any block. The justification for the name candelabra is shown in Fig. 1. The members of y are called branches, and S is called the sfem of the candelabra. These configurations were first introduced by Hanani (see Definition 2 of [8]) who used quite different terminology. The intersection size of two candelabra systems with the same points, branches, and stems (X, S, y, ql) and (X, S, y, q2) is defined to be the number of common blocks, i.e., 14,.n cd. Let JC( II, g :s) denote the set of possible intersection sizes of candelabra systems CQS(n, g : s). (Determination of the sets JC(n, g : s) is a generalization of the flower intersection problem for triple systems studied by Hoffman and Lindner [12].)

230

A. Hartman,

2. Yehudai

Fig. 1. A candelabra

The number

of blocks

in a candelabra

of type (n, g

system

:s). g : s) is

CQS(n,

We can construct a Steiner quadruple system of order ng + s-from a candelabra system CQS(n, g : s), a hole system HQS(g + s : s) and either an SQS(g + s) or an using the replacement property as follows. Let (X, S, y, p) be a sQs(s), CQS(n, g:s) and let (Gi U S, S, qi) be n hole systems HQS(g +s, s), for let (G, U S, r) be an SQS(g + s) and let (S, t) be an i= 1,2,. . . ) n. Further SQS(s). Now we can construct an SQS(ng + s), (X, b) either by letting n-l b=pUrUIJqi

b=pUtUfiqi.

orbyletting i=l

As above

Lemma

this gives us the following

i=l

results.

2.2. (a) JC(n, g : s) + J(g + s) + C:=;’ JH(g + s :s) c J(ng + s),

(b) .JC(n, g : s) + J(s) + CL, JH(g + s : s) c J(ng + s), A third technique for obtaining quadruple systems with different intersection sizes is the method which we call swapping the hole. Let (X, q) be an SQS(V) which contains two sub-configurations, a subdesign (Y, r) which is an SQS(u) (Z, S, t) which is an HQS(w :s), (i.e., Y c X and r c q) and a hole subdesign again, with Z CX and t c q. Suppose further, that Y fl Z = S, then we can construct a new SQS(v) with point set X and block set q’=(q\(rUt))Ur’Ut’, where (Y, S, r’) is an HQS(u :s) and (Z, t’) is an SQS(w). slightly, using the following structures.

We generalize

this idea

Intersections

of Steiner quadruple

systems

Let us call the system (X, Y, Z, S, p) a double-holed intersecting holes of orders u and w and double-hole

231

system of order v with of order s-denoted

DQS(u : u, w : s)-if: (1) (XI = v, ]Y( = U, (ZI = w, and ISI = s, (2) Y, Z =X7 (3) Y fl Z = S, and (4) p is a set of 4-subsets which is not a 3-subset of Y or Z is contained

of X, called

blocks,

of Y or Z is contained in any block.

such that

in a unique

every

block,

3-subset

of X

and no 3-subset

The number of blocks in a double-holed system DQS(V : u, w :s) is given by b, - b, - b, + b,. As above, we denote the set of possible intersection sizes of double-holed quadruple systems by JD(v : u, w : s), and constructing these systems in the obvious way from candelabra systems gives the following. Lemma 2.3.

JC(n, g : s) + c:Lf

Now we define

JH(g + s : S) c JD(ng

an entwined system

denoted

+ s : g + S, g + s : 3).

by EQS(u,

w :s) to be a system

(Y, Z, S, p) satisfying: (1) IYI = U, IZI = w, and ISI = s, (2) Y fl.Z =S, and (3) p is a set of 4-subsets of Y U Z, called blocks, such that every 3-subset of Y and every 3-subset of Z is contained in a unique block, and no other 3-subset of Y U Z is contained in any block. The number of blocks in an entwined system EQS(u, w :s) is given by b, + b, - b,. If JE(u, w :s) denotes the set of possible intersection sizes of entwined systems then, it is clear that the following generalizations of Lemmas 2.1 and 2.2 hold. Lemma 2.4. (a) JD( u : u, w : s) + JE(u, w : s) c J(v), (b) JC(n,g:s)+~;_;JH(g+s:s)+JE(g+s,g+s:s)cJ(ng+s).

and hence

One can construct entwined systems with different intersection Steiner systems and hole systems in a straightforward way, yielding Lemma 2.5. J(u) + JH( w : s) c JE(u,

sizes by using the following.

w : s) and JH(u : s) + J(w) c JE(u,

w : s).

On the other hand, one can swap the hole as follows. Let (Y, r) be an SQS(u), let (Y, S, r’) be an HQS(u :s), let (Z, S, t) be an HQS(w :s) with Y n Z = S, and let (Z, t’) be an SQS(w). Both (Y, Z, S, r U t) and (Y, Z, S, r’ U t’) are entwined systems, and their intersection is of cardinality jr n r’l + Ir n t’l + It n t’l, since r’ tl t must be empty. We use the formula to prove the following result: (Note that the result for JE(8,8: 4) is essentially due to Lo Faro [16].)

232

A. Hartman,

Lemma Proof.

2.6. (J(u) \ {b,}) + (J(w)\ {b,})

Z. Yehudai

c JE(u,

Let Y, Z and S be sets of cardinalities

w : 4). u, w and 4, with

Y II Z = S. Let

(Y, r) be an SQS(u) such that S $ r and let (Y, r’ U {S}) be another SQS(u). Now, since these two systems disagree on at least one block, we can choose them so that jr fl r’] takes any value in the set J(u)\ {b,}. Similarly, if (Z, t U {S}) and (Z, t’) are both SQS(w)s and S $ t’, then It fl t’l can be made to take any value in J(w)\

{b,}.

Furthermore

the result.

r n t’ is empty,

so the hole swapping

construction

yields

0

The above result, with u, w E (8, lo}, is particularly useful parity problems caused by the fact that both J(8) and J(10) numbers.

in overcoming contain only

the even

3. The case 10 mod 12 Since J(10) = (0, 2, 4, 6, 8, 12, 14, 30) is known, and Colbourn and Hartman [l] proved that if 0 E JH(22: 10) then J(V) = I(V) for all II = 10 (mod 12), with u 2 46, the logical place to begin is with a study of the sets J(22) and JH(22 : 10). We approach these problems via candelabra systems CQS(3,6: 4), and we begin by defining some sets of blocks used in the construction of such systems. The branches of the candelabra will be the sets G, = {a,: a E Z,} with i E Z,. The stem of the candelabra is the set S = {mO, w1, a2, 00~). Consider the following sets of blocks: q&cl, a

+

n1,

n2,

4

=

{l%

63,

bl,

c2):

a,

b,

c E z5,

b + c = ni (mod 6), i = 0, 1, 2, 3},

qj(j, k) = { {ai, (a + 3b),+l> (j - 2~ - 3b);+2, (k - 2~ - 3b);+,}: u~Z~,i~Z~,b=0, l}, qti(d)= i=Z3,

{{a;, (a +d);, b=O, l}.

(a +36);+1,

(a +3b

+d)i+~}:

a EZ6,

of a graph Now let F = (fo, fi, $2 b e an ordered triple of one-factors set Zg, and let M = [m(i, j)] be a 3 by 3 matrix with rows and columns members of Z3, and with the property that each row is a permutation define a fourth set of blocks by qzB(F, M) = {{a,, b;, ci+l, di+,): We shall use the following

two triples

{a, b) ~4, {c, d) •f,(,,~),

of one-factors:

g; = {{i, i + l}{i + 3, i + 4}{i + 2, i + 5)) i = 0, 1, 2, h,={{2i,2i+2}{2i+3,2i+5}{2i+1,2i+4}}

i=O,

1,2

with vertex indexed by of Z3. We

i, i E Z,>.

Intersections

and the matrix 120 M, =

120 [ 120

1

41=

4&

q3

=

29

systems

334) u q&,5)

CQS(3,6

: 4) using these, building

lJ q‘zs((& hi, &)I M,), u q*B((hOr hi, h2), M,).

that each of these qj is the block set of a CQS(3,6:4), (see [ll] construction). The intersection sizes which interest us are

lqi n q21 = 0, and lqi n q31 = 36, Now let iV, for each x E JP(3) + JP(3) + JP(3) = JP(9) with it4i in precisely x places, and form the block sets 4X0= G(2)

074, 3) u

4x1

%4,3)

=

blocks.

u qzA(2) u q2B((go, g1, g2), M),

4, 5, 0) u 93(1> 2) u &A(l) 4, 5, 3) u %(l, 2) u %4(l)

4409

It can be verified for the general

233

systems

.

We now form some candelabra

q2 = q-(3,

of Steiner quadruple

9mux

43(L

u 4s(l,

qx2 = 440, 2, 394) u

43K

be a matrix

5) u qzA(2) u %B((g”,

g1>

gd,

which agrees

Nxh

5) u qzA(2) u %B(@Ol g1, &I, w, 5) u &M(2) u qaB((g0,

g1,

gd,

N).

Again, each of these is the block set of a CQS(3,6:4) and jql cl qxjl = 9x + 72j + 72. Summarizing these constructions we have the following. (0, 36) U ((72, 144, 216) + 9JP(9))

Lemma 3.1. Now,

c JC(3,6

: 4).

since

J(10) = (0, 2, 4, 6, 8, 12, 14, 30) and hence (1, 3,5, 7, 11, 13, 291, we deduce, by Lemmas 2.5 and 2.6 that T(23) U ((25) This directly

implies

+ T(4)) U ((29)

+ J(10)) c JE(lO,

by Lemma

=

10:4).

that

((1) + T(51)) U ((54) + T(4)) U ((58) +J(lO)) Hence,

5(10:4)

2.4(b)

c JE(lO,

10:4) +J(10:4).

we have the following.

Lemma 3.2. ((1) + T(348)) U ((351)

+ T(4)) U ((355) +J(lO))

cJ(22).

The values in Z(22) not covered by the previous lemma are 0, and 385 - t for f E (8, 12, 14, 15, 17, 19, 20, 21, 23, 25, 27, 29, 35). We shall first show that 0 E J(22) and also show that 0 E 5(22: 10) by completing some of the CQS(3,6 : 4) designs to obtain SQS(22) and HQS(22: 10) designs. We shall use the following shorthand for the blocks of an SQS(l0) and an HQS(l0 : 4). The notation

234

A. Hartman,

will denote

2. Yehudai

the set of blocks

((9

a(O, i), a(I,

U{{m,

a(&

i), a(29 i)>: i E Z,>

0), a(i +j,

l), a(i + 2j, 2)): i, j E Z,)

U {{a(& j), a(i, j + l), a(i + 1, j), a(i + 1, j + l)}: i, j E Z,} U {{u(i, j), u(i, j + l), u(i + 1, j + 2), a(i + 2, j + 2)): i, j E Z,}. The notation a(O, 0) a(I,

0)

a(2, 0)

42, 1) 42, 2) [ml x

[ will

denote

the

1

40, 1) u(O,2) a(l, 1) a(172)

same

set

of

blocks

with

the

{m, ~(0, 0), ~(1, 0), ~(2, 0)}, thus forming the block consider the following pair of HQS(22: 10) designs:

the

block

set of an HQS(10:4).

exception

of

Now

h,=~,U[p,~~x~~]U[:)~~~x~~],

h~=~~U[~~,~~x~~]U[-:,.~x~~].

Now, we have h, n h, = {{cy, w2, 01, 2,}, {mi, w2, 02, 22}, }. If we form h2’ by permuting the points 0, and O2 and fixing the remaining 20 points of the design, we have h, fl h,. = 0 and thus 0 E JH(22 : 10). This fact, by Colbourn and Hartman’s construction, is sufficient to prove the following. Theorem

3.2. J(v) = Z(v) for all ZJ= 10 (mod 12), with II 2 46.

Furthermore,

since 0 E J(lO),

and by Lemma

2.1 we have

0 E 5(22).

(3.3)

We now construct an SQS(22) with three subsystems, two of order 8 (which intersect in a block) and one of order 10 which intersects one of the subsystems of order 8 in a block. As above, the point set of a the system is S U {ai: a E Z,, i E Z,>. fi = ((0, I>, {2,3}, Let fo= {{0,3), {I, 41, (2,511, (1, 2}, (3, 4}}, and let Mid be the matrix with mij = j. Let r, be the block set of an SQS(l0) with point set S U contains the block {mO, mi, 00, 3,). Let rl be the block set point set S U {a,: a E Z,}, hole S, and which contains the

(4, S}},

f2= {{0,5},

{a,: a E Z,} and which of an HQS(10:4) with blocks {wO, cq, Oi, 3,)

235

Intersections of Steiner quadruple systems

with point set {cQ~, ml, lr, 4,). Let r, be the block set of an HQS(10:4) S U {u2: a E Z,}, hole S, and which contains the blocks {ccl,,,ml, 02, 3*}, and {%, 9, 22, 52). The final set of 144 blocks is defined by the following four Latin squares of side 6, indexed by Z,+ For each cell in Latin square Li with column index j, row index k, and entry X, let q_(Lo, L1, L2, Lx) be the set of all blocks of the form

and

0

2

13

5

3

4.

1

4

0

2

10

2

5.

543210

210543

152403

405132 L1=

Lo=

354021

021354

543210

230541

132405

405132

230541

4

321054

105432

5

3.

230541

014325 L, =

Lz=

105432

543210

452103

014325

543210

321054

Now form the block set q=(Lo, LI, Lz, L3) lJ 43(1,5) U q2A(2) U qzB((.I& fi, f2),

Mid)

U

ro U

ri

U

r,.

This forms the block set of an SQS(22) with a subsystem r,, of order 10 on the points S U {ao: a E Z,}, a subsystem of order 8 on the points {m,, ml, Oo, 30, Or, 3r, 02, 3*}, and a further subsystem of order 8 on the points 1~0, O”1, 00, 30, 11, 41, Deleting the two O,, 3,) shows {MO>O”lJ bzz - 2bs + b4 E JD(22 Now, computing as

22, 52). pairs of subsystems which intersect on the block that 342 = bz2 - blo - b8 + b4 E JD(22: 10,8: 4), and 358 = : 8,8 : 4). above, we have

T(21) U (25, 27) U ((29) +J(8)) c JE(lO, 8:4)

and

(3.4)

T(8) u (11, 12) u ((13) +J(8)) c JE(8,8:4). Now, applying Lemma 2.4(a), we obtain (342) + (T(21) U {25,27} U ((29) + J(8))) cJ(22) (358) + (T(8) U (11, 12) U ((13) +5(8))) cJ(22).

and (3.5)

236

These

A. Hartman,

results

show that

Z. Yehudai

1(22)\ (368) cJ(22).

We now construct

order 22 with precisely 368 blocks in common. usual, and define the following ingredients. Let s,, be the block set of an SQS(lO) contains defined

the block

{m,,, lo, 2,, 5,).

two systems

We use the same

with point

Let the matrix

point

of

set as

set S U {a,,: a E Z,} and which M’ and Latin

squares

,!,I be

by 0

1

2

[ 0 0

21

21

1 ,

M’=

432105

543210

543210 L;,=

L;=

321054

321054

435102 L; =

105432

210543

210543

054321

054321

102435

354021

021354

432105

210543

210543 Lj =

102435

021354

354021

105432

543210

543210

435102

Now form the block set

qm(L&L;, G> G) U 441, 2) U qu(l) This system

contains

the 17 blocks

U

qm((hor h,, hJ, M') U sg u rl u r,.

listed below.

{9,> 10, 01, 32>{mo1 10, 31, O,>{~“> 20, 11, 32}{% in qp(G, (9,

G,

I%, G),

20, 01, 32){9>

in qm(-G, G,

50, 11, O,}

G,

20, 31, O*>{m,, 50, 01, Oz}{~,, 50, 31, 32) G),

{lo, 2”, Or, 0,){10,

2”, 31, 3J

{lo, 50, Or, 3,){2,,

5”, lr, 3,1{1,,

in q,(l,

2))

5”, 02, 32~{11, 3r, 02, 3*)

in q2B((h0, hr, h2), M’), {%, 10, 20, 5&{%,

Y, 01, 31}{%,

9,

02, 32)

in s,U rl U r2.

Intersections

Now form a new system following

of Steiner quadruple

by removing

these

systems

17 blocks

231

and repalcing

them

by the

list of 17 blocks.

The resulting system blocks, thus proving

intersects

the previous

system

in precisely

385 - 17 = 368

368 E 5(22) Combining Theorem

results

(3.6) (3.2),

(3.3),

(3.5),

and (3.6), we obtain

the following.

3.7. 5(22) = Z(22).

We now consider the last remaining value u = 34 in this section. As before, we begin by defining some sets of blocks used in the constructions of candelabra systems CQS(3,lO: 4). The branches of the candelabra will be the sets Gi = {ui: a E ZIO} with i E Z,. The stem of the candelabra is the set S = {a%, tQ1, w2, 9). Consider the following

sets of blocks:

{{mi,

q&no,no, n2, no) = u + b +

c E ni

~0,

b17~2):~3 6, c E Zloy

(mod lo), i = 0, 1, 2, 3},

43(no, nl, n2) = {{ai, (a + l)i, bi+r, ci+2}: a, b, c E Zro, a+b+csni(modlO),i=O,

1,2}.

Now let F = {fo, fi, . . . , f6} b e an ordered one-factorization of the graph with vertex set Zlo, and edge set E = {{i, i +j}: i E Zlo, j = 2, 3, 4, 5). (Such a one-factorization exists by a result of Stern and Lenz [20].) Let M = [m(i, j)] be a 3 by 7 matrix with the property that each row is a permutation of Z7. We define a third set of blocks by

q2(F,M) = ({a;, bi,

Ci+l,

di+l>: {a,b) ~6, {C, d) Efm(i,j), i

E

Let Mi be the matrix, each of whose rows is the identity permutation let IV, for each x E JP(7) + JP(7) + JP(7) = JP(21) be a matrix which

Z3, j

E

Z,}.

on Z7, and agrees with

A. Hartman.

238

Z. Yehudai

ikfi in precisely x places. Now form the block CQS(3, 10 : 4) using these building blocks. 41 = qm(0, 1,2,

sets of some

candelabra

systems

3) U q3(4, 6, 8) U qz(F, M),

qox = q&3> 0, 1, 2) U &8,

4, 6) U q@,

Nx),

qlx = qm(O> 1, 2, 3) U q3(8, 4, 6) U q#,

Nx),

q2r = qm(0, 1, 2, 3) U q3(4, 6, 8) U q,(F,

A$).

Each of these qti is the block set of a CQS(3,10:4), as is q1 (see [9] for the general construction), and lqlflqo,l=25x, ~q,flq1*_~=400+25x, and lq,n q&l = 700 + 25x. Th’IS means that {0,400,700} + 25JP(21) c JC(3,lO: 4). Lo Faro [17] showed that T(47) U (91) cJ(14) and thus T(46) U (90) c JH(14:4). Now it is easy to verify that T(227) cJ(14) + JH(14:4) + JH(14:4), and by Lemma 2.2(a) these two facts imply that T(1452) c3(34). To obtain the higher intersection numbers, we note that Hanani [8] has constructed an SQS(34) with a subdesign of order 16. Thus 1356 = bs4 - b16 E JH(34: 16). Now invoking Lemma 2.1 we have (1356) +5(16) c5(34). Lo Faro [18] proved that 5(16) = 1(16) and hence we can deduce that J(34) = Z(34). Summarizing the results of this section, we have the following. Theorem 3.8. For all v = 10 (mod 12) with J(10) = (0, 2, 4, 6, 8, 12, 14, 30).

TV2 22 we have J(v) = Z(v),

and

4. The cases 14 and 38 mod 72 Etzion and Hartman Hartman showed that if Puccio’s results on 5(14) therefore, by Lemma 2.2, ‘For v = 14 or 38 (mod SQS(v). Let v = 12n + 2, an SQS(n + 1). We now

[3] showed that 0 E JC(3,12:2), and Colbourn and T(144) c 5(38) then J(38) = Z(38). Using Lo Faro and [19] we have T(144) cJ(14) +J(14:2) +5(14:2) and we have 5(38) = 1(38). 72), with v 2 86 we use the following construction for then n = 1 or 3 (mod 6), and n 3 7. Let (X G {co}, q) be construct a CQS(n, 12: 2) with point set (X X Z12) U S {x} X Z12, x E X, and stem S. The blocks are constructed

where ISI = 2, branches as follows. For each block {x, y, z, m} E q which contains 00, construct a CQS(3,12 : 2) with branches {x} x Z12, {y} X Z12, {z} X Z12, and stem S. For each block {x, y, z, t} E q which does not contain 00, construct 123 blocks, one for each cell of a Latin cube of side 12 with rows {x} X Z12, columns {t} x Z12_ The blocks consist of levels {z} X Z12, and symbols {Y> x z12, quadruples of the form {i, j, k, m} where m is the symbol in row i column j and level k.

Intersections

A Latin

of Steiner quadruple

cube of side 12 can be constructed

systems

from eight disjoint

239

Latin

side 6. Using Lemma 4.1 of [l] concerning the possible intersection disjoint Latin cubes of side 6, we have the existence of a pair of Latin side 12 with precisely

entries

of

sizes of cubes of

for all

123 - 14) U { 123 - 12, 123 - 8, 123}.

k E T(

The number b n+l

k common

cubes

of Latin -

n(n

cubes used in the construction 1)/6 = n(n - l)(n

-

is

- 3)/24,

: 2) we have

and since 0 E JC(3,12 n(n-l)(n-3)124 c

i=l Now, by Lemma

12 : 2).

T(123 - 14) c JC(n,

2.2, and the fact that 0 E 5(14),

we have

n(n-l)(n-3)/24

T(123 - 14) = T(1714n(n

c

- l)(n

- 3)/24) c.Z(l2n

+ 2).

i=l By Colbourn and Hartman it is sufficient to show that T(12n(12n - 12)/6) c J(12n + 2) to imply that J(12n + 2) = Z(12n + 2). But 1714n(n - l)(n - 3)/24 > 12n(12n - 12)/6 for all n L 4, therefore we have proved the following. Theorem

4.1.

For all Y = 14 or 38 (mod 72), with v 2 38, J(V) = Z(v).

5. The case 8 mod 18 Etzion and Hartman [3] have shown that 0 E JC(3, n : 2) for all n = 2 (mod 6). Now Lo Faro [17] has shown that .Z(n + 2) = Z(n + 2) for all II + 2 = 4 (mod 12), with n + 2 2 16. In Section 3, we showed that Z(n + 2) =.Z(n + 2) for all n+2=10(mod12), withn+2 2 22. If we now apply Lemma 2.2 we can deduce that T(3b,+, - 14) cJ(3n + 2) for all n = 2 (mod 6), with 122 14. Now, by the results of Colbourn and Hartman, it is sufficient to show that T(3n(3n

- 12)/6) c .Z(3n + 2)

for all n = 2 (mod 6),

with II 2 14, and since 3b,+, - 14 > 3n(3n the following. Theorem

- 12)/6 for all n 2 10, we have proved

5.1. J(v) = Z(V) for all u = 8 (mod 18), with u 2 44.

6. The case 2 mod 24 Etzion and Hartman [3] have shown and, Colbourn and Hartman [l] have

that 0 E JC(4, n : 2) for all II = 0 (mod 2), shown that it is sufficient to prove that

240

A. Hartman,

2. Yehudai

T((v - 2)(v - 14)/6) cJ(v) t o complete the determination of J(v) for all v = 2 (mod 24), with u 3 50. Since we use an induction argument, we begin with the case v = 26, and an investigation of the set JC(4,6:2). We begin by defining some sets of blocks used in the constructions of candelabra

systems

CQS(4,6

: 2). The branches

of the candelabra

Gi = {a,: a E Z,} with i E Zq. The stem of the candelabra Consider the following sets of blocks:

will be the sets

is the set S = {mO, ml}.

: a0~ bl, c2}: a + b + c = 2j (mod 6) (a, b, c) = (0, 0, 0) (mod 2))

qrn(i, j) = { {mi, U {{y,

a”,

b,, c2}: a + b + c = 2j + 1 (mod 6) (a, b, c) = (1, 1, 1) (mod 2))

U { {“i7

uO~

bl, ~,}:u+b+c=2j+2(mod6)(u,

6, c)=(l,

0, l)(mod2)}

U { {mi, a,,

b,, c3}: a + b + c = 2j + 1 (mod 6) (a, b, c) = (0, 1, 0) (mod 2))

U

bz, cj}: a + b + c = 2j (mod 6) (a, b, c) = (0, 1, 1) (mod 2))

{{O”i,

uO~

u ((9,

ao, b,, q}:

u

{{O”i9

Ol,

ba, c3}: a + b + c = 2j + 2 (mod 6) (a, 6, c) = (1, 0, 1) (mod 2))

u

{{O”i,

u19

b2, c3}: a + b + c = 2j + 1 (mod 6) (a, b, c) = (0, 1, 0) (mod 2))

u

{{O”1-i3

uO~

a + b + c = 2j + 1 (mod 6) (a, b, c) = (1, 0, 0) (mod 2))

bl, cz}: a +b +c=2j(mod6)(u,

b, c)=(l,

1, 0) (mod2))

U { {m1_i, uo,

b,, c2}: a + b + c = 2j + 1 (mod 6) (a, b, c) = (0, 0, 1) (mod 2))

u { {O”1-i2

uO,

bl, c3): a + b + c = 2j + 2 (mod 6) (a, 6, c) = (0, 1, 1) (mod 2))

U

uO,

bl, cg}: a + b + c -2j

{{031-i~

+ 1 (mod 6) (a, b, c) = (1, 0, 0) (mod 2))

u {{T-i,

ao, b2, cj}: a + b + c = 2j (mod 6) (a, b, c) = (0, 0, 0) (mod 2))

U { {mll-i7

uO3

b2, cj}: a + b + c = 2j + 1 (mod 6) (a, b, c) = (1, 1, 1) (mod 2))

U

{{ml-i,

ul9

bz, cg}: a + b + c = 2j + 2 (mod 6) (a, b, c) = (1, 1, 0) (mod 2))

U

{{ml-i,

ul,

b2, cj}: a + b + c = 2j + 1 (mod 6) (a, b, c) = (0, 0, 1) (mod 2)).

Now

let Gi={{2i+2,2i+4},

one-factors

with vertices

{2i+3,2i+5}}

for i=O,

1,2

be three

partial

in Z6, and define

q&j) = {{a(),bl, x2, y2}: a = b (mod 2) a + b + 2c = 2j (mod 6) (x, y} E G,} u {{a,,

b,,x,,y,}:

a S b (mod2)u

+ b +2c=2j+

U {{a,,

b3, x0, yo}: a = b (mod 2) a + b + 2c = 2j (mod 6) (x, y) E G,}

1 (mod6)

(x,y}

E GC}

u I{%, b3,.r,,y,}:ufb(mod2)u+b+2c-2j+l(mod6){x,y}EG,.}. Now let Ho= ((0, I>, (2, 31, (4, 5))) HI = ((0, 31, (1, 41, (2, 511, Hz= ((0, 5}, (1, 2}, (3, 4}, be three one-factors with vertices in Z6. Furthermore let M = [m(i, j)] be a 4 by 3 matrix, with each row being a permutation of (0, 1,2}

241

Intersectionsof Steiner quadruplesystems

and define qH(M) = {{a,,, bl, x2, y2}: a f b (mod 2) a + b + 2~ z 1 (mod 6) {x, y> E H,,o, u {{a,,,

b,, x3, y3}: a = b

lJ

{{U2?

b3, x0, y”}:

u

{{U2>

b3, x,,

.J

(mod 2) a + b + 2c = 0 (mod 6) {x, y} E H,,,,,,)

a f b (mod 2) a + b + 2c = 1 (mod 6) {x, y} E Hm~2,c~}

y,}: a = b (mod 2) a + b + 2c = 0 (mod 6) {x, y} E ~IL~~,~,}.

let C, D, E, F be ordered one-factorizations of Kh with vertex set Zg. Say each ci is a one-factor, and similarly D = (di), c = (co, Cl, . . . , c4) where with each row being a E = (e,), F = (5). Let N = [n(i, j)] b e a 2 by 5 matrix, permutation of (0, 1, 2, 3, 4) and define:

Finally,

qF(C,

D, 6

F, N) = {{a,,

u

{I@29

bo, ~1, Y,): b2,

~3,

{a, b) E c; {x, y>

~+,,,;,O~i~4}

Yx): {a, b} ~ei (~9 y> ~f,(t,;,Os

i <4}.

We now define

some specific one-factorizations, and matrices of permutations to be used in constructing various candelabra systems CQS(4,6 : 2). Define the matrix AI, to be the 4 by 3 matrix with each row equal to [0 1 21, and define the matrices M, for x E JP(3) + JP(3) + JP(3) + JP(3) = JP(12) to be matrices with exactly x entries in common with M,. Similarly, define the matrix N, to be the 2 by 5 matrix with both rows equal to [0 1 2 3 41, and define the matrices N, for x E JP(5) + JP(5) = JP(10) to be matrices with exactly x entries in common with Ni. We give an explicit definition of Ns for use in a specific construction.

Let 4

0

N = 01234’

1

2

Let F. and Fl be the following Fo= (((0,

3

1 ordered

11, (2, 31, (4, 5}}, ((0,

((0, 41, (1, 3], {2,5>>,

Fl = (((0, 11, (2,411 {3,5]], ((0, 51, {1,2],

{3>4],

Now we define the following these building blocks. ql=

of Kg.

one-factorizations 21, {1,4},

(3, 51, ((0, 31, (1, 51, 12, 4))

{{0,5],

{1,2],

{3,4]]),

{{0,4],

{1,5],

{2,3]],

{{0,3],

{L4],

systems

CQS(4,6

U qF(F;,, F;,, I;;,, F,, Ni),

qzry = 4m(O, 1) u q&L) U q&K)

U q&T,,

F,, 81, F,, NY),

q3xY= qm(l, 0) U q&O) U q&t,&)

U q,(F,>

81, F,, 8, NY),

q4xy = qm(O, 0) U q&O) U q&K)

U qF(F;,, F,, F,, I;;,, NY),

4s = q-(0,

1) U qc(l)

U q&‘%)

U qF(F,,

1,

{2>5]]).

block sets of candelabra

qm(O* 0) U qc(O) U q,(M)

{{0,2>1 (1, 31, (475)

&I, F,, F,, Ns).

: 2) using

242

A. Hartman,

Z. Yehudai

Each of these qj is the block set of a CQS(4,6: construction) with intersection sizes as follows:

2) (see

[3, 61 for the general

q2ryl= 18x+ 9y,

141 n

141 r-l q3xyl = 144 + l&c + 9Y, lql n q+l=

288 + 18x + 9y,

141 fl q51 = 3. Therefore,

we have proved

Lemma 6.1.

the following.

((0, 144, 288) + 18JP(12)

+ 9JP(lO)) U (3) c JC(4,6:2).

Now, by Lemma 2.2, and the fact that Z(8) +J(8) i = 21, 22, 24, 28}, we can deduce the following. Lemma 6.2. 1(26)\({1} 37, 43, 45)) cZ(26).

U (650 -t:

+3(8)

= (2i: 0~ i < 18,

t = 15, 16, 17, 18, 19, 21, 23, 25, 29, 31, 33,

Hartman [lo] constructed on SQS(26) with subsystems of orders 10 and 8 which intersect in a block the same system also contains two subsystems of order 8 which intersect in a block. The first pair of subsystems are block-disjoint from an additional subsystem of order 10, thus we have {577}+J(lO)cJD(26:10,8:4)

and

623~JD(26:8,8:4).

since 577 = bz6 - 2blo - b8 + b4, and since 623 = bz6 - 2b8 + b+ By (3.4) we have T(21) U (25, 27) U ((29)

+5(8))

c JE(10,8:4),

and T(8) u (11, 12) U ((13) Now, applying Lemma 2.4(a), { 1,633) c J(26). We now show that 1 E J(26). {{uco,

ui+l9

u1+29

(which form an SQS(8)); { {ucn,

ui+3,

ui+5,

(which form an SQS(8) 41, =

41

u

{[Th

%+4){4,

+J(8))

c JE(8,8:4).

we can strengthen

Lemma

Let [a,,, a,, . . . , u6, a,] denote ui+3,

%+5,

%+6):

disjoint w 1,

ai+i,

ai+2,

%+a>:

Z(26)\

the set of blocks

i E G>,

and let (uo, a,, . . . , u6, am) denote ai+6}{ui7

6.2 to obtain

i E G>z

from the first). Now let

Oj, li, . . . , 5i]: i = 0, 1, 2, 3},

q5, = q5 U { ( mO, aI, Oi, li, . . . , 5,): i = 0, 1, 2, 3).

the set of blocks

Intersections

of Steiner quadruple

243

systems

These two systems intersect in precisely three blocks. Applying the permutation (O,O,) to the system q5, yields a further system, and the reader may verify that the intersection of this system with ql, is only one block. Thus we have proved the following. Theorem

6.3. Z(26) \ (633) c J(26) c Z(26).

We now proceed by induction to prove the next theorem. Theorem

6.4. For all v = 2 (mod 24) with v 2 50, we have J(v) = Z(v).

Proof. Let

v = 24n + 2 with it 3 2. By Colbourn and Hartman’s result it is sufficient to prove that T(4n(24n - 12)) c J(24n + 2). Now Etzion and Hartman proved that 0 E CQS(4,6n, 2) for all it 2 1, so by Lemma 2.2(b) it is sufficient to show that T(4n(24n - 12)) c i

J(6n + 2:2) = f: J(6n + 2)

i=l

for all II 3 2.

i=l

When n = 2 the result follows from Lo Faro and Puccio’s result that T(47) U (91) cJ(14). When IZis odd, the result follows from Lo Faro’s result that J(6n + 2) = Z(6n + 2) since 6n + 2 = 8 (mod 12). When 12= 4 the result follows from Theorem 6.3. When IZ= 2 or 6 (mod 12) with n 3 6, the result follows from Theorem 4.1. When II = 10 (mod 12) the result follows from Theorem 5.1. Finally, when n = 0 (mod 4) with n 2 8, the result follows from the induction 0 hypothesis of this theorem.

7. Conclusions

Summarizing all the results described above we have the following. Theorem 7.1. For all u = 2 or 4 (mod 6) we have J(v) = Z(v), with the exceptions of v = 10 and 14, and the possible exception of v = 26. In these cases we have

J(10) = (0, 2, 4, 6, 8, 12, 14, 30}, Z-(47) U {49,51, 53, 54, 55, 56, 59, 61, 91} c J(14) c T(75) u {91}, Z(26)\ (633) c J(26) c Z(26). We have no doubt that the doubtful case with v = 26 will eventually be settled. We conjecture that J(26) = Z(26). The final determination of J(14) is a very difficult computational problem-it seems reasonable to believe that, say T(65) c J(14), but we do not even hazard a conjecture concerning the values between 65 and 75.

244

A. Hartman,

2. Yehudai

Many other intersection problems have been raised that the most interesting of these is the determination

by this paper. We believe of the sets JH(v : w). It is

not even known, in many cases, whether b,:, = b,, - b, E JH(v: w), since this problem is equivalent to the embedding problem for Steiner quadruple systems (see [6] for recent wS8 we have

progress

on this problem).

We conjecture

that for v 3 2w and

JH(v : w) = T(b,:, - 14) U {b,:, - 12, b,,:, - 8, b,:,} with only finitely

many

exceptions.

References PI C.J. Colbourn

and A. Hartman, Intersections and supports of quadruple systems, Ann. Discrete Math., to appear. D.G. Hoffman and C.C. Lindner, Intersections of S(2,4, u) designs, Ars PI C.J. Colbourn, Combin. 4 (1991) 182-185. [31 T. Etzion and A. Hartman, Towards a large set of Steiner quadruple systems, SIAM J. Discrete Math., to appear. of Steiner quadruple systems having a prescribed [41 M. Gionfriddo and C.C. Lindner, Construction number of blocks in common, Discrete Math. 34 (1981) 31-42. PI M. Gionfriddo and G. Lo Faro, On Steiner systems S(3, 4, 14), Ars Combin. 21 (1986) 179-187. PI A. Granville and A. Hartman, Subdesigns in Steiner quadruple systems, J. Combin. Theory Ser. A 56 (1991) 239-270. [71 H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960) 145-157. Canad. J. Math. 15 (1963) 702-722. PI H. Hanani, On some tactical configurations, 191 A. Hartman, Tripling quadruple systems. Ars Combin. 10 (1980) 255-309. systems containing AG(3,2), Discrete Math. 39 (1982) 293-299. [lOI A. Hartman, Quadruple for quadruple systems, J. Combin. Theory Ser. A [Ill A. Hartman, A general recursive construction 33 (1982) 121-134. problem for Steiner triple systems, 1121 D.G. Hoffman and C.C. Lindner, The flower intersection Ann. Discrete Math. 34 (1987) 243-248. among Steiner systems, J. Combin. Theory Ser. A [I31 E.S. Kramer and D.M. Mesner, Intersections 16 (1974) 272-285. number of triples in iI41 C.C. Lindner and A. Rosa, Steiner triple systems with a prescribed common, Canad. J. Math. 27 (1975) 116661175; corrigendum, Canad. J. Math. 30 (1978) 896. system-a survey, Discrete Math. 22 (1978) 1151 C.C. Lindner and A. Rosa, Steiner quadruple 147-181. systems of order v = 7.2” with n 3 2, Ars [161 G. Lo Faro, On the set J(u) for Steiner quadruple Combin. 17 (1984) 39-47. systems, Ann. Discrete Math. 30 (1986) u71 G. Lo Faro, On block sharing Steiner quadruple 297-302. systems having a prescribed number of blocks in common, 1181 G. Lo Faro, Steiner quadruple Discrete Math. 58 (1986) 167-174. 1191 G. Lo Faro and L. Puccio, Sull’insieme J(14) per sistemi di quaterne di Steiner, Boll. Act. Gioenia Sci. Nat. 17 (1984) 221-237. another proof of the PO1 G. Stern and H. Lenz, Steiner triple systems with given subspaces; Doyen-Wilson theorem, Boll. Un. Mat. Ital. A (5) 17 (1980) 109-114.