3. MORE PROPERTIE S OF ADMISSIBLE ARRAYS 13

It follows from Theorem 3.1 that

j = l T3

j = 1

rJ

and this contradiction proves the corollary. •

A weaker version of Corollary 3.3 plays an important role in the computer

program that we use to find the array admissible sets (see Chapter 6).

COROLLARY

3.4. Let S = {pi | 1 i ra +1} be a collection ofm + 1 distinct

positive integers pi with 1 pi n. Assume thatpi+pj n for 1 i j m + 1.

Assume that Sj C S, 1 j [i are pairwise disjoint sets with S = Uj=i &j-

F°r

each j , 1 j ii, assume that there is an integer rj 1 such that

gcd(p, q)\rj for all p G Sj and all q ^ p,q G S.

Let Sj = | Sj |. //

£?! (3-17)

then S is not an array admissible set for n.

PROOF .

Let S = { i G Z | l i r a + l} with the usual ordering and suppose

that E is a set with |E| — n. Assume, by way of contradiction, that S is array-

admissible for n. Then there exists an admissible array { ^ : Z — E | Z G S ' } and

a permutation a of S such that Oi has minimal period pi := pa(i)- If we define

Bi = {Oi(j) | j G Z}, the fact that pa^) + pa(j) n£oriijm + l implies

that Bi n Bj ^ 0 for 1 i j m + 1. We define Sj by

Sj = {a-1(i)\pieSj},

so \Sj\ = Sj, and observe that our hypotheses imply that if i G Sj then

gcd(pi-Upi)\rj a n d gcdG5i,j3i+i)|?

Theorem 3.1 now implies

3 = 1 T

which is a contradiction. •

Corollary 3.4 motivates the following definition.

DEFINITION 3.5. A set S C {1,2,... , n} satisfies the generalized condition C

for the integer n if S does not contain disjoint subsets Q and R with the following

properties:

1. gcd(a, /?) = 1 for all a G Q and /? G Q U i? with a ^ /?.

2. a + /3 n* := n - Y,^eQ 7

f o r a11 a,P

^ R

w i t n

« 7^ /?•

3. there exist pairwise disjoint sets Sj C R, 1 j // with Uj=1Sj = R and

such that for each j , 1 j /x, there is an integer rj 1 such that

gcd(p, q)\rj for all p G Sj and all q ^ p, # G i?.

4. E ^ = i ^ l , where Sj = \Sj\.