CHAPTER 2
Basic properties of admissible arrays
In this chapter we collect some properties of admissible arrays proved in [13].
We start with some definitions.
DEFINITION 2.1. A set 5 C {1,2,... ,n} satisfies condition A for the integer
n if S does not contain a subset Q such that
1. gcd(a, (3) = 1 for all a, (3 G Q with a ^ /?;
2- EaeQan
For example, the set {8,9,16,18,25} does not satisfy condition A for the inte-
gers n with 24 n 50, as one sees by taking Q {9,16,25}.
DEFINITION 2.2. A set S C {1,2,... ,n} satisfies condition B for the integer
n if S does not contain disjoint subsets Q and R with the following properties:
1. gcd(a, (3) = 1 for all a e Q and (3 e Q U R with a^ (3.
2. i? has r + 1 elements, r 1, and gcd(o;, /?)|r for all a, f3 E R with a ^ (3.
3. For all a,f3 e R with a / f t t t + ^ n * : = n - Sae Q
a
-
Note that condition A is contained in condition B. An example of a set that
does not satisfy condition B for n equal to 34 is {16,26, 34}.
DEFINITION 2.3. A set S C {1,2,... , n} satisfies condition C for the integer
n if S does not contain disjoint subsets Q and R with the following properties:
1. gcd(a, (3) = 1 for all a e Q and (3 e Q U R with a ^ f3.
2. there are integers r\ 1 and ri 1 such that gcd(a,/?)|r, r = r\T2, for all
a,f3 e R with a ^ /?,
3. there exists jo e R such that gcd(a,7o)|ri for all a e R with a ^ 70 and
4. |i?| rir
2
- r2 + 2 and a + [3 n* := n - ^
a G Q
a for all a, (3 e R with
An example of a set that does not satisfy condition C for n equal to 27 is
{12,16, 20, 22}. In this case Q = 0, i? = {12,16,20, 22}, n = r2 = 2 and 70 = 22.
DEFINITION 2.4. A set S C {1,2,... ,n} satisfies condition D for the integer
n if S does not contain a set R with the following properties:
1. \R\ = m + r 1, where m 2 and r 2, and gcd(a, /3)|r for all a,[3 e R
with a 7^ /?.
2. there exist disjoint subsets R\ and R2 of i? with R\ U i?2 = -R, |i?i| = m
and |i?2| = r 1, X^aei? a n, and a + /? n for all a G i? and (3 £ R2.
An example of a set that does not satisfy condition D for n equal to 21 is
S = {9,15,16,21}. In this case R = {9,15,16,21}, m = 2 and r = 3.
DEFINITION 2.5. A set S C {1,2,... ,n} satisfies condition E for the integer
n if S does not contain disjoint subsets Q and R with the following properties:
1. gcd(a, (3) = 1 for all a e Q and (3 e Q U i? with a ^ /?.
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