A theorem on independence

Discrete Mathematics North-Holland

287

120 (1993) 287-289

Note

A theorem on independence Daniel Q. Naiman Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, USA

Henry P. Wynn City University. London, UK Received

21 May 1991

Abstract Suppose we are given a family of sets collection of sets H,(j,), . . . . H,(j,+,) inclusion, then the maximal size of an collection of half-spaces H, , , Hk in

W= {S(j), jgJ}, where S(j)= n;=, Hi( j), and suppose each has a lower bound under the partial ordering defined by independent subcollection of ‘Z is k. For example, for a fixed W’, we define V to be the collection of all sets of the form

where xi, i = 1, _. , k are points in [Wd.Then the maximal size of an independent collection of such sets us k. This leads to a proof of the bound of 2d due to Renyi et al. (1951) for the maximum size of an independent family of rectangles in [W“with sides parallel to the coordinate axes, and to a bound of d + 1 for the maximum size of an independent family of simplices in [W”with sides parallel to given hyperplanes H, , , H,, 1.

1. Introduction A finite collection JZ?= {&i, i = 1, . . , n} of sets is said to be independent if every set of the form (-)l= r Xi, where each Xi is either Ai or its complement A;, is nonempty. Independence for sets arises in a variety of settings, some of which are described in Naiman and Wynn [3]. The problem of finding upper bounds for the size of an independent collection of sets with certain geometric properties goes back at least as far as to Renyi et. al. [S], who give upper bounds on the maximum possible number of Correspondence to: Daniel Q. Naiman, University, Baltimore, MD 21218, USA. 0012-365X/93/$06.00

0

1993-Elsevier

Department

of Mathematical

Science Publishers

Sciences,

B.V. All rights reserved

The Johns

Hopkins

288

D.Q. Naiman, H.P. Wynn

independent sets of a given type in IWd.They show among other results that the maximum number of independent rectangles (with sides parallel to the coordinate axes) in [Wdis 2d, and the maximum is d+l. Naiman independent

and Wynn collection

M(d)=L3d/2

number

of independent

balls of equal radius in IWd

[4] showed that the maximum number of rectangles in an in [Wdall of which are translates of one another is given by

J.

2. The main theorem We generalize the result for rectangles in Renyi et al. [S] to other sets which are formed by intersecting translates of half-spaces. First we describe an abstract result in which geometry is not mentioned at all. Fix a family of sets %‘= {S(j), jEJ>, where each set S(j) can be represented in the form S(j)=

A

Hi(j)

i=l

for some (doubly-indexed)

family of sets

{H,(j),i=l,...,

k,jEJ}.

Theorem. Suppose every collection of k + 1 sets Hi( jl), . . . , Hi( j,+ r ) has a lower bound under the partial ordering dejined by inclusion, for i= 1, . . . , k. Then the maximum number of sets in an independent subcollection of 9? is k. . , S(k + 1) be sets in %?,so that

Proof. Let S(l), S(j)=

h

Hi(j),

j=l,...,

k+l.

i=l

For fixed i, we can fix an index ji such that Hi( ji) is a lower bound Hi(j),j=l,..., k+l, that is, j=l,...,

Hi(ji)GHi(j), Theset

{ji,i=l,...,k} JE{l,

consists

k+l,

and we have

Since J#j(i),

for i=l,...,k

k.

of at most k elements,

. ..) k+l}-{j(i),i=l,...,

Hi(ji)GHi(J),

i=l,...,

i= 1, . . . . k. we have

k},

for each of the sets

so we can fix

A throrem

289

on independence

Thus

and the collection

cannot

be independent.

0

Corollary 1. Suppose H 1, . . , H, are half-spaces the collection S=

in Rd and %7= %?{H 1, . . , Hk } denotes

of sets of the form

f’ xi+Hi, i=l

where x 1, . . . , xk are points subcollection of % is k.

in lWd.Then the maximum number of sets in an independent

Indeed, for each i the xi + Hi are totally ordered by inclusion, so the lower bound assumption is trivially satisfied. Based on the results Renyi et. al., Griinbaum [l] conjectured that the bound of d + 1 for the maximum size of an independent collection holds whenever the collection consists of homothetic copies of a given compact convex set. This conjecture fails for translates of a given rectangle by the result in Naiman and Wynn [3]. Nevertheless, we are able to use Corollary 1 to obtain Griinbaum’s conjectured bound for simplices with parallel bounding hyperplanes. Corollary 2. The maximum number of sets in an independent family of d-dimensional simplices in Rd having parallel bounding hyperplanes is d + 1. As another following

simple

application

of Corollary

1 we obtain

another

proof

of the

result.

Corollary 3 (Renyi et. al. [S]). The maximum number of sets in an independent family of hypercubes in Rd with sides parallel to the coordinate axes is 2d.

References [II B. Griinbaum, Venn diagrams PI E. Marczewski, Independance 122-132. c31 D.Q. Naiman and H.P. to Griinbaum, Discrete c41 D.Q. Naiman and H.P. tube-like problems via c51 A. Renyi, C. Renyi and a n dimensions, Colloq.

and independent families of sets, Math. Mag. 48 (1975) 12223. d’ensembles et prolongements de mesures, Colloq. Math. 1 (1947)

Wynn, Independent collections of translates of rectangles and a conjecture due Comput. Geom. 9 (1993) 101-105. Wynn, Inclusion-exclusion-Bonferroni identities and inequalities for discrete Euler characteristics, Ann. Statist. 20 (1992) 43-76. J. Suranyi, Sur I’independence des domaines simples dans I’espace euclidien Math. 2 (1951) 130-135.