3. MOR E PROPERTIE S O F ADMISSIBLE ARRAYS 17

s£-fc+i r vfr

n

r , _ , | _ ,

+ 1

a n d t o

G P

( [

T/jL — k ^ r jji—k

for 1 j 7^_fc. If we define

S^-fc = S^-k U {gM_fc+1,... , ZM!lfc+i}

then gfc+i = lcm (S^-^) and it suffices to analyze lcm(5M_/e). Since

we have

I-S'/x—fcI 7/x-fc-i^-fc-

Proceeding as before, we can divide S^-fc into 7AX_/C_i disjoint subsets of size at

most l^-k- So

7/u-fc-i

Sn-k = \j Sn-kj and |5(M_fc)j| l^-h-

By definition, for every 5 € Sfa-^j, 1 3 7/z-fc-i

rM_.fc|5 and - ^ - G

P ( [ - ^ - ] ) .

So an application of rule B yields, for 1 j 7M_/C_i,

——-lcm(S

( M

_*

w

) = Z„-*lcm(- Si/A-k)j) e p(l^k[-^-])

I li—k — 1 ' / L A — k x ' ii—k '

and

If we define QJrL_k\ = lcm(Sr(Al_fc)j), 1 j 7M_fc-i, the claim follows. This proves

the theorem. •

From the results that we will prove in this paper, it actually follows that for

n 33 a set S C {1,2,... ,n} is array-admissible for n if and only if S sat-

isfies conditions generalized C, D and E for n. If n equals 34 the following set

{12,14,16, 20,34} satisfies the generalized condition C and condition D and E but

is not array-admissible (see condition G). In the second part of this chapter we

derive more conditions from the admissible arrays.

The second assumption in the generalized condition C states that the sum of

any two elements in R is bigger than n*. This condition is needed to insure that the

sets Bp, p G R, have pairwise nonempty intersection. When n* (n) becomes larger

the assumption that the sum of any two elements in R is bigger than n* becomes

rather strong. The following corollaries are simple consequences of Theorem 3.1

and give other conditions.

We first need an auxiliary lemma.