Abstract
Let
X = {
x
1,
x
2,…} be a finite set and associate to every
x
i
a real number
α
i
. Let
f(
n) [
g (
n)] be the least value such that given any family
F
of subsets of
X having maximum degree
n [cardinality
n], one can find integers
α
i
,
i=1,2,… so that
α
i
−
α
i
|<1 and
∑
x
i ϵ E
a
i−
∑
x
i ϵ E
α
i
≤ƒ(n)
∑
x
i ϵ E
a
i−
∑
x
i ϵ E
α
i
≤
g(n)
for all
E ϵ
F
. We prove
f(n)≤n − 1 and g(n)≤c(n
log n)
1
2
.