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DISCRETE

MATHEMATICS

ELSEVIER Discrete Mathematics 137 (1995) 345 349

Note

On the density of sets of divisors

R.E.L. Aldred a'*, R.P. Anstee b' 1

Department o[ Mathematics and Statistics, University qf" Otago, P.O. Box 56, Dunedin, New Zealand

b Department o/' Mathematics, University qf British Columbia, # 121-1984 Mathematics Road,

Vancouver, BC, Canada V6T 1 Y4

Received 7 July 1991

Abstract

Consider the lattice of divisors of n, [1, n]. For any downset (ideal) J in [1, n] we get

a forbidden configuration theorem of the type that if a set of divisors D avoids certain

configurations, then ] D ] ~< I J ]. If we let 5 ¢ be the set of minimal elements of [ 1, n] not in J, then

we forbid in D the configurations C(s) (defined in the paper) for sE5 e. This generalizes a result of

Alon and in turn generalizes a result of Sauer, Perles and Shelah.

Keywords: Extremal set theory

1. Introduction

Let [m]--{1,2,

the divisors are in one to one correspondence with the vectors in [ka+l] ×

[k2 + 1 ] × ... × [kr, + 1]. For a divisor s = I]i~mlp~i ', we will also use the vector notation

s=(sl, s2 ..... s,,). Let [-1, n] denote the lattice of divisors of n. We will be discussing

subsets D of [1, hi. These subsets can be interpreted in various ways. We could view

D as submultisets of a multiset (where element i occurs at most k~ times), each

submultiset corresponding to a divisor. Alternatively, D yields a (kl+l)×

(]£2+ 1)X ... ×(k,~+l) (0, 1)-matrix in m dimensions with each 1 corresponding to

a divisor. Also if we let an (m; kl, k2, ..., k,,)-matrix A be a matrix on m rows whose

entries in row i belong to {0, 1, 2 ..... k~ }, then if the columns are distinct, each column

corresponds to a divisor. We find the divisor notation somewhat easier to use.

,m}. For n having a prime factorization n=pklp k .... k~

"'" Pm

1 Support provided by NSERC. Research done during a sabbatical at University of Otago.

* Corresponding author.

0012-365X/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSD1 0012-365X(93tE01 14-J

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R.E.L. AIdred, R.P. A nstee / Discrete Mathematics 137 (1995) 345-349

A downset J (ideal) in [1, n] is a subset so that ifasJ and bla then bsJ. Let J be

any downset in [1, n] and let S~ be the set of minimal elements not in J. We follow the

shifting idea in [1]. Frankl [4] has popularized the shift operator in extremal set

theory. Let a shift operator T act on subsets of [1, n] so that for a subset D of [1, n]

(i) ]T(D)I=]DI,

(ii) T(D) is a downset,

(iii) T(D) does not contain [1, s] for sos p.

We may conclude that T(D) ~_ J and so [DI -%< I J I. We must define a specific T and

determine the properties of D so that (i)-(iii) hold. We will require that D have no

configuration C(s) for s~ which we now define informally. Imagine [1, n] as a box of

integer grid points in ~m, with axes xl, x2 ..... xm. Let s = (Sl, s2 ..... sin). For m = 3, C(s)

corresponds to s3 + 1 planes parallel to x 3 =0, each plane containing s2 + 1 lines each

parallel to x2=x3=0 and on each line there are sl + 1 distinct points/divisors. For

arbitrary m, C(s) corresponds to sm+l (m-l)-dimensional subspaces parallel to

xm=0, each (m-1)-dimensional subspace containing Sm-l+l (m-2)-dimensional

subspaces parallel to x~_ 1 = x~ = 0, etc., and in each one-dimensinal subspace (paral-

lel to x2 =x3 ..... Xm=0) there are sl + 1 points/divisors.

Let us state Alon's result [1] to see how we are generalizing it. Let 5'~A (subscript

A for Alon) be a family of subsets of [m]. Let A be a matrix on m rows and for S ~ [m],

let Ais be the submatrix of A consisting of those rows of A indexed by S.

Theorem 1 (Alon[1]). Let A be an (m; kl, k2 ..... km)-matrix with n distinct columns.

Assume for no S6SaA that Ais contains I~i~s(ki+ 1) distinct columns. Then if we let

5'~={ s~[l' hI: s=l-[ pk' s°me S ~ Sf A } '

and o¢ be the downset in [1, n] for which 5, ~ is the set of minimal elements not in J, then

n~<l~l.

Note that ~ and hence J have a special structure, nonetheless, our forbidden

configurations C(s) for se5 e correspond to the same restriction on A. Our result

(Theorem 5) extends Theorem 1 (which in turn generalizes a basic results of Sauer [5]

and Perles and Shelah [6]) to an arbitrary ideal. Some other examples where J has

special structure and better forbidden configurations can be determined are explored

in [3].

2. Main results

We define T via the shift operator of Alon. Let

Ti j(al, a2 ..... ai ..... am)=~ "(al'

•

1

am)

a2

a i --

((al,a2,...,a,...,am)

if ai =j,

otherwise.

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We then let

Ti'j(a)={ Tri'j(a) a

if otherwise, Ti'i(a) ~ D'

and extend naturally to define T~,j(D).

We define

Ti=(Ti, 1 ° Ti,2 ..... L,k,) ..... (Ti, 1 ° Ti,2)° Ti, 1,

i.e. shift as much as possible in the ith coordinate. Finally the shift operator T is

defined as follows:

T= Tmo Tm_l ..... T2o T1.

Lemma 2. Let D be a subset of [1, n]. Then T(D) is a downset.

The proof is postponed until later in the paper.

It is useful to talk about certain projections of sets of divisors. For J_~ [m], let

O,j= { j~s p'JJ: d=(dl, d2 ..... dm)eD }.

Now for any e=(el ..... em)eD let

Lemma 3 (Alon [ 1]). For each J ~_ [m], I TI,j(D) IJ [ <<- I D I IJ I with equality for J = [m].

Under our shift operator we can determine which configurations in D give rise

to [1, s] in T(D), for s=i],~EmlP~i'~Sf. Define the configuration C(s) of divisors in

D inductively as follows. Let s'= 1-]i~[,,-ij P~'. The configuration C(s) consists of s,.-4- 1

different values of 0~<il ~i2 ~ ""ism+l ~<k,, so that, defining it--(0, ... ,0, it), in each

D[ ~m-ii,i, we have the configuration C(s').

Lemma 4. If D has no configuration C(s) for seSP, then neither does T(D).

We delay the proof until later in the paper.

We can now state our main result.

Theorem 5. Let J be a downset in the lattice of divisors [1, n] and let 5 ~ be the set of

minimal elements of[l, n] not in d. Let D be a set of divisors of n with no configuration

C(s) for sE5 '~. Then

IDI~I~I.

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R.E.L. Aldred, R.P. Anstee/Discrete Mathematics 137 (1995) 345 349

Proofi By Lemma 2, T(D) is a downset. By Lemma 3 (or directly), I T(D)I=[D[. By

Lemma 4, T(D) does not contain [1, s] for so5 p since D does not have C(s). So

T(D)~_d and the result follows. []

Because we have obtained a result for any downset d, configuration theorems

follow in profusion. We can for example obtain results which yield a bound on D that

is polynomial in m such as the following result of Woodall.

Theorem 6 (Woodall, see Problem 10.10 [2]). Let D be a set of divisors of n such that

for x, y~D and z any product of I prime factors (not necessarily distinct),

then

z does not divide gcd(x, y);

+(kl+k2+'"+k'+m).l +""

O)

(2)

Proof. The configuration C(s) implies the existence of numbers with large common

factors. Let Trim(s)= s/pl, where i= rain {j: p~ divides s}. Now, if D contains C(s) then

D contains two divisors x, y with Trim(s) a divisor of gcd(x, y). Now apply

Theorem 5 with d being all possible products in [1, n] with exactly l factors and so

5 ~ is all possible products in [1, n] with at most l+ 1 factors. Thus our condition (1) on

D implies that D has no C(s) for s~5 °. []

We can interpret Theorem 5 in terms of an (m; kl + 1, k2 + 1 ..... k,,+ 1)-matrix A. If

A has n distinct columns and no configuration C(s) for so5 e, then n ~< [d I. But what is

C(s)? Let s=plpzp3 for example. Then C(s) corresponds to a row and column

permutation of any 3 x 8 submatrix in rows 1, 2, 3 of A of the form

a

e

g

a'

e

g

b

e' e'

g

b' c

f

g' g' g' g'

c'

f

d d'\

f' f'),

g

(3)

where the restrictions on the entries are a ~ a', b # b', .... g # g'. At this point we do not

have any applications of our general result in this setting but we note that, for

example, if we require any square submatrix of A to have determinant in { - 1, 0, 1 },

then A has no configuration C(plp2p3), since the entries of(3) are forced to be 0, 1 and

so we get a 3 x 3 submatrix of determinant _+ 2.

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3. Proofs of lemmas

Let us now return to fill in the proofs deferred earlier.

Proof of Lemma 2. We use induction on m to show that if D= Tio Ti-l ...... TI(D),

then /31~1,~ is a downset for each possible e. We define b= T~_I ..... Ts(D), and

assume that/:31t,_ll.e is a downset for each e and show that T~,i(b)IE,_~3,, is a downset

for each e.

Let a=(al ..... ai ..... am)ET(D) and let a* be such that Tij(a*)=a,

(al ..... a* ..... am).

Case 0: a* ~j. The result follows by induction.

Case 1:

i=j and ai=j. Now Tij(a)=a so (al,az,...,j--I .... ,am)~D. /)]Ei-I],~,

e=(0 ..... 0,j--l, ... ,am), contains [1, I]l~ti 1]P~'] and as such is fixed by Tij. Hence

Tij(l))]1i_ ll, ~ contains [-1, FIl~ti_ llP'Z'].

Case

2: a*=j, ai=j--1.

By the inductive

[-1, I]l~[i-11P'l']. For each a'=(a'l, a~ ..... a'i 1,J, ai+l .... ,a,,)~f) and so a"~Tij(b) or

Tij(a')=a". Thus Ti.i(/3)[Ei_I],, contains [1, I]~ti_~lpf~]. Now, TI(/))=/), and Ti is

a composition of T~j operators. Hence, inductively, D[ E~,~-~],~ is a downset for each e.

But (a~ ..... al ..... am)~f) implies (a~, ..., a~ ..... am)~D for 0 ~ a; ~ ai (by the properties

of Ti, and since TI(/)) = D) which implies (a'~ ..... a~ ..... a,,)~D for all 1 ~a~i~a~, for all

1 ~j~i, by properties of downsets. So Tio Ti-~ ..... T~ (D)[t~,~ is a downset for every

e. Now the result follows by induction when i=m.

a*=

a* "

^

hypothesis, D]~-ll,~* contains

[]

Proof of Lemma 4. The proof uses induction on m. Assume that T(D) has [1, s] for

s - [Ii, tml P i so p,, 1-I i~1,. - a I P i ~ T (D) for 0 ~< k ~< s,,. Now if we let D' =

then T(D)=T,.(D). Thus there exist io<il<'"<ism

O<<,k<<,Sm. But now, using Lemma 1, DIEm_~3,e is a downset and so, if we let

s -1-[i~tm-llP~, then for each x~[1, s'], we have xp~D'

induction to get that for each of the s,.+ 1 values ik, Olt~-zl,** contains C(s') and so

D contains C(s'). []

__ Si k $i

T i o Ti - 1 ° TI (D),

ljPl ~D for

s,

•

so that P~ 1-I~1,.- ,

p__ si

for O<~k<~s,,. Now apply

References

[1] N. Alon, On the density of sets of vectors, Discrete Math. 46 (1983) 199-202.

[2] I. Anderson, Combinatorics of Finite Sets (Oxford University Press, New York, 1987).

[3] R.P. Anstee, Maximum number of l's in a (0, 1)-matrix subject to forbidden configurations, preprint.

[4] P. Frankl, On the traces of finite sets, J. Combin. Theory Ser. A 34 (1983) 4145.

[5] N. Sauer, On the density of families of sets, J. Combin. Theory Ser. A 13 (1972) 145-147.

1-6] S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages,

Pacific J. Math. 41 (1972) 247-261.