COLLOQUIA
4.
MATHEMATICA
COMBINATORIAL
THEORY
SOCIETATIS
AND ITS APPLICATIONS,
On the set of non pairwise
of a number
JANOS
BALATONFORED
coprime
BOLYAI
(HUNGARY),
1969
divisors
bY
P. Erdds
J. Schijnheim
and
Calgary, Canada
Budapest, Hungary
1. INTRODUCTION
The problems
the idea of considering
generalizing
entities
intersection
more general
It is amusing
number
theorems
for finite
to observe
of prime
factors
identities
let N have
ti,4-cd2+...+tis
are,
for example;
should hold for
the divisors
of a given
5: ( N I- representing
t prime
where
factors,
of N- satisfies
py’ py’
a large
coefficients.
n denotes
number
In order
as its decomposition
. pzs
integer.
the number
(redurdantly)
of
the
of identities
to give a simple
in primes,
then
= n,
f
S;(N)
i=O
3
from
sets and of
that the symbol
on binomial
tot
:: 0‘,=0(,‘...=5:
L
theorems
arise
than sets.
of N containing
generalizing
example
about in ths lecture
them in the sense that the obtained
Such entitites
divisors
we will be talking
=1
then
=
%x,+l).
L=:
Sk(N)
=::j
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and (0) generabzes
2n f”)
i i, = 2*.
t/=3
The first
by De Bruijn,
Sperner’s
non-trivial
Van Tengbergen
theorem.
consequences
prefer
and Kruijswijk,
to consider
the numbertheoretical
and is a generalization
various
of
sn)
[l]
concepts.
and
to 5 {CL,)
We might
mention
is not always
new facts make their
language
and for its immediate
more general
the results
5 = (L,,X2’...’
framework
has been obtained
theorems
[l]
of
which are
of the same theorem.
for its simplicity
containing
in this direction
One of us [2] generalised
Although
attractive
result
applications,
Katona
[3]
[2] using integral
5a1,)
proved
valued
and
[2] is
other
authors
theorems
functions
on
.
here,
that a theorem
a translation
appearance.
used in [l]
in this more general
of the corresponding
For example
theorem
if n different
for sets,
sets
A, I A,, -. .
differences
Ai - Aj is >- n ,
‘“> A” are given, then the number of different
but if n different
divisors
d,,d, , . , d, of a positive integer are given then
the number
above,
of different
ratios
is not necessarily
( d;, dj>/d;
these correspond
to the differences
Z n . See [4].
2. RESULTS
We will
generalise
it and reveal
a’.
and
If
A;n Aj # #
l-q = p-’
now turn our attention
the new facts involved.
sets
are distinct
A,,A,,...,A,
for every
i,, j
then
m C-2”-’
intersection
theorem
The considered
result
[5],
of [5]
sets such that ~A,uA~LJ,...~A,I=~
and for every
n
there
are
satisfy
the
such sets.
b’ . Moreover,
requirements
to another
if the sets
of a’ then there
8, , B, , _. , BP,,-1
exist sets
also satisfy
The more general
B, , B, , . . . , B,,
Bs+, , B,,,
5.~ 2n-’
, .“. , Bzn-4
such that the
these requirements.
problem
in the spirit
of this talk is the following:
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of the introductory
words
is:
a*. To determine
are different
then
divisors
m 5 f(N)
are,
Moreover,
for every
replaced
to b* is negative,
by another
also satisfy
G,, G,, . .. , G+(N)
In order
value
in the following
to have an affirmative
g(hl)
divisor
>I,
such divisors.
then there
of a* and m < f CMl
such that
in b’
= f(N)
N,m
The bound f( N ) is determined
the answer
D, , D, , . . . , D,
is it true that if G, , G, , . . . , G,
the requirements
NG s+l ““‘GflN)
fC N 1 such that if
of N, and each two of the D’ s have a common
and there
b”.
satisfying
a number
which is best possible
are divisors
are divisors
of
of
these requirements?
theorem,
whereas
answer
f(N)
is
but is not very
illuminating.
THEOREM
1.
If D,,...,Dm
decomposition
common
in prime
divisor
(1)
where
then,
may
m
the summation
best possible,
subset
is
divisors
of an integer N whose
t
and each two of the D’s have a
“, pTi
=
denoting
=
is over
for every
all subsets
the product
N there
{ i, ,... , $1
is considered
are f(N)
of
{ 4, . . . , t f
to be one. The result
divisor
is
no two of which are
prime.
Denote
fL-A)
factors
>i
and for the empty
relatively
are different
where
by F;(t)
a family
the intersection
of 2t-’
of any two
subsets
Aj ’ s is non-empty.
be the set of all these families.
15i5S
Trivially
.
-
Let
of
Fi
(t)
,
As far as we know the value
of s is not known.
- 371
Aj, 15 j I 2t-i
Let
N = t,
p:’
.
g(N)
=
Put
Ecj
(2)
where the minimum
summation
over
THEOREM
2.
are relatively
prime.
divisors
,...j
G,,
v rxi,
V=l
all the
Aj ‘S
be m distinct
divisors
Assume
m c g(N),
1 L_i c s
F; ,
all the 2t-’
G,, . ,G,
G,,,
z
is to be taken over
is extended
Let
min
F!t)
L
Fi ,
Of
and the
Aj = pi,,...,
~~~ .
of N no two of which
Then there are
g(N)
-m
further
so that no two of the g(N) distinct divisors
Gg(N)
are relatively
prime. Theorem 2 is best possible.
lc-isg(N)
The proof
utilizes
the following
lemma.
LEMMA
If
positive
integers
P,rP2,..”
M; depending
c h,,
are two different
are different
Pt
partitions
h,,...
only on p;
, hJ
( i = 1,2,
u {k,,k,,...
of the index set
and
- 372
-
and cx, , ti2 , . . , CX~ are
primes
. . . , t)
and if
, k,,j
ij,2,.
.. ) t 3
such that
then
the greatest
is greater
common
divisor
Of p;, pi, . . . pi
and
p,.,, ph, . . . p,,
P
than 1.
d
3. PROOFS:
Proof
of
the
Lemma.
Suppose the contrary;
Pj, Pj,
... Pi,,
then
ph, phZ’-’
Ph, is a proper
divisor
of
hence
and (i) contradicts
(ii).
PROOF OF THE THEOREM
Denote
by M the square
and by 5 a set of divisors
divisor
>l,
as required
free part of N i.e.
of N, any two members
M = p, pz . . . pt
of S having
a common
in a?
Let
(Pi,
(3)
Pi
P
be a factorization
contain
‘**
integers
of M into two relatively
composed
from
prime
pi, , . . ) pi
ti
hence contains
and from
P
P
at most
that factorization.
To show
factors,
max 4 T o( i, ) c(/ T tii, )
v=l
3-I
This shows ISI L fINI I
] S] z f(N)
consider
(4)
and say i, = 4 if equality
holds
2 Lx/
p. , . . . . p.
JI
members
all such factorizations
which
P
lT q,
i=l
then S cannot
P
lTo[~,
v=l
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,
JP
deriving
from
(3) of M for
Put in S all the
rtii,
integers
v=l
In this way we construct
prime
{by our Lemma)
a set of divisors
and containing
members,
where
restricted
to index subsets
every
the second summation
satisfying
N, which completes
the proof
ipj’IJ
G’s are relatively
of
prime.
FCt)
say to
Fjt’
is
divisors
for
beany
m<glN)
m
Put
Thus any two of the m sets
intersection,
hence they all
a But then one can clearly
at least (the summation
G,,...,G,
Fct’)
p
prime.
the first
1.
G,,...,G,,
have a non-empry
ilj5m
belong to the same
set
Let
while
leads to f(N)
of Theorem
of N no two of which are relatively
No two of the
ii ,,..,,
is as in (l),
(4). This
NowweproveTheorem2.
divisors
of N no two of which are relatively
is extended
over the
add to the
2t-’
sets
Aj
.
m zgCN)-m
further
divisors
no two of which are relatively
To show that Theorem
prime,
2.is best possible
- 374
-
which proves
observe
Theorem
that if F, gives
2,
the minimum
relatively
in (2) and we consider the gr N) divisors
(no two of which are
pj B,
15
j 5 pt-’
prime)
z, pi,
,
11
po’cq,;
corresponding
relatively
to the 2t-i
prime
A’s of F, clearly
to at least one of the given
Unfortunately
(2) is not a very
every
g(N)
useful
further
divisor
of
N
divisors.
and illuminating
formula.
is easy to bee that
g(N)2 d-1 +@llin{
where
in (5) the summation
+Li,,
a/f&l
i=l
i,=l
is extended
over
except the empty set and its complement
Assume
CX, 2 , . I z tit ,
all the sunsets
t)
{I,...,
perhaps
t-i
(6)
F, would then consist
g(N)
= tit
y,hi+l)
5
of all the sets containing
- 375 -
is
t .
.
of
{ 1, . . . , t j
It
REFERENCES
[l]
DE BRUIJN, N.G., VANE.
TENGBERGEN,
On the set of divisors
of a number.
23 (1949-51),
191-193.
[2]
SCHONHEIM,
J., A generalization
of results of P. ERD&,
and D. KLEITMAN
concerning
Sperner’ s theorem.
Theory (to appear}
[3]
KATONA,
G., A generalization
of some generalizations
Theorem.
J. Combinatorial
Theory (to appear).
[4] MARICA,
J. and SCHijNI-IEIM,
GRAHAM.
Can. Math.
C.A.,
and KRUIJSWIJK, D.
Nieuw Arch. Wiskunde (2),
G. KATONA
J. Combinatorial
of Sperner’
J. Differences
of sets and a problem
Bull. (to appear)
[S] P. ERD&,
CHAO KO and R, RADO, Intersection
theorems
finite sets, Quart. J. Math. 12 (1961), 313-320.
- 376 -
s
of
for systems
of