3. MORE PROPERTIE S OF ADMISSIBLE ARRAYS 19

COROLLARY

3.9. Assume that S = {pi,P2,P3iP4iP5}i where 1 pi n for

1 i 5 and pi ^ pj for 1 i j 5. Assume that

1- Pi + P5 n for i ^ 5, Pi+p4 n for i ^ 4 and px + p2 + £3 n.

2. gcd(p^,p5) = 2 /or 1 i 4 and gcd(p^,pj)|4 /or 1 i j 4.

Then S is not array-admissible for n.

PROOF.

Because p\+P2+Pz ^, Lemma 3.8 yields that some subcollection

{h.k} C {1,2,3}, ix ^ 12, satisfies BPii n BPi2 ^ 0. Let L = {pi^Pi^p^Pb} and

suppose that there is a permutation a such that qi = pa(j) is the period of the

ith

row of an admissible array on n symbols {6i : Z — E | 1 i 4}. As in Remark

3.2, let Bqi = {0i(j) I j G Z}. Then BpnBq^0 for all p, g G L. If 5i = {p5} with

ri = 2 and $2 = {PinPi2,P4} with r2 = 4, then the conditions of Corollary 3.3 are

satisfied and

n r2 2 4

Thus Corollary 3.3 shows that 5 is not array-admissible for n. •

Corollary 3.9 motivates the following definition.

DEFINITION 3.10. A set S C {1,2,... , n} satisfies condition F for the integer

n if there does not exist a subset {pi,P2,P3,P4,P5}, where 1 pi n for 1 i 5,

Pi y^ Pjfor 1 i j 5 and

1. pi + p5 n for i ^ 5, p^ + P4 n for i ^ 4 and p\ + P2 + Pz n.

2. gcd(p;,p5) = 2 for 1 i 4 and gcd(pi,pj)|4 for 1 i j 4.

An example of a set S C {1,2,... ,36} that does satisfy the conditions gen-

eralized C, D and E but does not satisfies condition F (and hence is not an array

admissible set for n = 36) is given by {12,16, 20,28,34}.

COROLLARY

3.11. Assume that S = {pi,P2,P3,P4,P5} C E = {1,2,... ,n}

?

where Pi ^ pj for 1 i j 5. Assume that

1. pi + P5 n for 2 ^ 5 and that Pi +P2 +Pz 2(n — P4).

2. gcd(p;,p5) = 2 /or 1 i 4, gcd(p*,p4) = 2 /or 1 i 3 and gcd(pi,pj)|4

/or 1 z j 3.

T/ien S is not array-admissible for n.

PROOF.

For 1 i 3, consider L\ = {pi,P4,P5}. If S is array-admissible,

L\ is array-admissible. Our assumptions imply that gcd(p, q) = 2 if p,q e Li

and p ^ q. Also BP4 n BP5 ^ 0 and £

P i

n BP5 ^ 0, i = 1,2,3. Thus condition

generalized C with S\ = Li and n = 2 gives a contradiction unless BPi C\ BP4 = 0

for i = 1, 2,3. Therefore we can assume that ((Jf=i BPi) H £

P 4

= 0. It follows that

BPi C Ei := T,-BP4 for 1 i 3 and |Ei| = n-p4. Since pi +P2+P3 2(n-p

4

)

we conclude from Lemma 3.8 that

BP1 H BP2 H 5

P 3

^ 0.

If L2 = {pi,P25P3P5} we conclude (because p^ -f ps n for 1 i 4) that

Bpn Bq ^ 0 for all p, q G L2. We also know that gcd(p^ps) = 2 for 1 i 3 and

gcd(p,g)|4 for all p,g G L2, p 7^ Q- If ft = {p5} with rx = 2 and 5

2

= {pi,P2,P3

with r2 = 4, the conditions of Corollary 3.3 are satisfied and

£i + f i i .

ri r2