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.
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