qL
ISRAEL JOURNAL OF MATHEMATICS. Vol.
51.
\os. l-2.
lo85
FAMILIES OF FINITE SETS
IN WHICH NO SET IS COVERED
BY THE UNION OF r OTHERS'
P. ERDOS,'P. FRANKL" AND Z. FUREDI'
"Mathematical Institute of the Hungarian Academy of Science,
oCNRS, 15 Quai A. France,75007 Paris, France
1364 Budapest, P.O.B. 127, Hungary; and
of an t2-set satisfying the
Let f,(n,k) denote the maximum
that
condition in the title. It is proved"iitTlliiu-."bsets
f,(n,r(t-
1)+
d\
k d\
-,,-\( " -r l/\I ( -,')
- at=
1+
fornsufficientlylarge
whenever d:0, 1 or d< rl2t'with equality holding iff there exists a Steiner
system S(t, r(t - I)+ 1, n - d). The determination of /.(n,2r) led us to a new
generalization of BIBD (Definition 2.4). Exponential lower and upper bounds
are obtained for the case if we do not Dut size restrictions on the members of the
family.
1. Preliminaries
Let X be an n-element set. For an integer k,0<k í n we denote by (f) the
collection of all the k-subsets of X, while 2' denotes the power set of X. A
family of subsets of X is just a subset of 2*.It is called k-uniform if it is a subset
of (f). A Steiner System 9 = S(t,k,n)is an 9 C(f) such that for every T€(í)
there is exactly one B e 9 with T CB. obviously, Ig|: (DlÖ holds. A g CG)
is called a(t,k,n)-packing iÍ.lP n P'|< t holds for every pair P,P,e g. V. Rödl
[10] proved that
(1)
max{19
|:
Q isa(t,k,n)-packing}
= (1
-
o(1))
/n\ //k\
I \'; )
\'; )
holds for all fixed k, Í wheneveÍ n..>@.
t This technical report is published as a result of an FCAC Foundation
Received April 30, 1984 and in revised form October 1, 1984
grant.
P. ERDOS ET
80
AL
Isr. J. Math.
Let fal (lb.l)
denote the smallest (greatest) integer (not) exceeding
respectively. We will use the Stirling formula, i.e., n! - (n le):/2rrn.
2.
a
(b),
UnÍform r-cover.free families
9 r-couer-free ít FoÉ.F'U...U4 holds for all
g.
(F,lFj iÍ ilj.) Let uS denote by Í,(n,k) the maximum
Fo,F,,...,F,e
cardinalityof an r-cover-freefamily g C(t),lXl: r. Letusset t: fklrl.Then
We call the family of sets
PnoposntoN
2.1.
(Dl(), a Í,fu, /c) (l)/(Í 1').
=
To prove the lower bound we show that there exists a (t, k, n)-packing of this
size. A (t,r(t-l)+l,n)-packing 3 is r-cover-free because IPnP' l=t-t
holds for all P,P'e 9. Generally
Exeuu-s2.2.
Let X=Yl)D,
lDl:d,
lYl=n-d
(t,r(t-I)*I,n -d)-packing over Y. Define gF:{D
u
and g
P:P e?}.
a
This example and (1) gives the lower bound in the following theorem.
THponBlt
(2)
2.3. Let k:r(t-I)+1+d
(r-o(r))
where
0<d<r.
Then for
(";o) I (;o)=,,r,.k)=(';o) I
n>nr(k)
(;')
holds in the following cases:
(a) d
:0,1,
(b) d < rl(Zt'),
(c)t:2andd<lzrl3l.
Moreouer, equality holds
in (2) iff a Steiner-system S(t,k - d,n - d)
exists.
This theorem determines asymptotically f,(n, k) for.several values of r and k.
r:3, k:6. The obvious conjecture that the
maximum I has the structure given by Example 2.2 is not true (cf. Theorem
2.6). A subset ACFe 9F is cal|ed an own subset of F if AÉF,holds for all
The first uncovered case is
F# F',e g.
Let us suppose X:{L,2,...,n} and define maxF:max{i: i e F}.
family gc(f), t, r>_2, is called a near t-packing iÍ
lF n F'l< t holds for all distinct F,F' e 9, moreover, lF ft F'l: t implies
maxFt.F,(in words: the Í-subsets of F containing maxF are own subsets).
DprINIroN
2.4.
PnopostrloN
A
2.5. IÍ g
C (f)
's
a near t-packing then 9 is r-cotser.free.
Vol.51.
19fJ5
NO SET IS COVERED BY r OTHERS
Pnoop. Suppose FCFr U...U
8l
F,e g. Since lFnf, l< r, the sets Fr-tF,
form a partition into Í-Subsets of F. Choose fi containing max F. Then F í-l f, is
a /-subsetof FcontainingmaxFandFn E C F. However, F aFi wassupposed
to be an own subset of E, a
THeonsu
o(n') edges.
2.6.
F,,
contradiction.
There exists
a near
2-packing
n
gC(;,) with (n'l(4r-2))-
This theorem and Proposition 2.1 give that f,(n,2r):
is easy to see that
PnoposnoN 2.7. For fixed k and
|imf,(n.D
exists whenever n
(1 +
o(1))n'111r -21. tt
r,
/ l n\,,'
(n):ti'SupÍ,,'
tn'k)/\,):c'(k)
/l \r/
--> cn.
By Proposition 2.1 and (2) we have
tl(k
t'
|'/\t-tl
l(k-t\
lo)=c.(k)=
, '
In Chapter 5 we get the slightly better
c,(k)-(k-dt)ttf
\
-])
r-1l
o
but we have no general conjecture for the value of c.(k) not covered by
Theorems 2.3 and 2.6.
3.
r-Cover-free famÍlies without size restriction
Denote by Í,@) the maximum cardinality of an r-cover-free family
lXl:n.
3.1. 0+ Il4r,)" < Í,@)< e(|+o(1'))nlr.
RsNaanr. In the case r:1 the constraints reduce to
9 C2,.
Trrponplt
well-known Sperner-property. Hence (see [11])
f
Suppose now that
/n\
,(n): \
friU ).
n is not too large compared to
r.
Fr,É-Fr, i.e., the
P. ERDOS ET
82
AL
Isr. J. Math
3.2. Let q be the greatest prime power with q
Y:
=\G.Let
x
graphs
be
the
underlying
polynomials
GF(q ) GF(q )
set and consider the
of the
of degree at most d over the finite field GF(q). Set
Ex,q.upt-p
9rl :
I{(x,s(x)): x e GF(q)}: c(x)
Then I F nF' l<
d
trotds
family.
:
ao* etx+
"'
+ e4xd,
a;e GF(q)}.
for F, F'e T*a, thus it is a l(q -1)ld)-cover-free
This yields the lower bound for 2r' < n in the following:
THponev 3.3. For
r: r\i
we haue
(I - o(\){nlll.]+l < Í,fu)=
For n
( ('j')
PnoposntoN
,|2le21
.
we have the foliowing easy
3.4. IÍ, <(!')
then
f,(n):
n.
4. Proof of Proposition 2.1
|f F is a maximal (t, k, n)-packing
such that
then for every G e G) there is an F €
I holds. Hence we have
lG n Fl>
(;)=á|{"=(f)
:
IGnnl=r}
|=,*l(ÍX
9
i-)
Using
(:\(k):/"\í1-,\
\k/\r/ \r/\k-tl'
this yields the lower bound.
For the proof of the upper bound let us define the family /[(F) the non own
parts of F with respect to 9, i.e.,
J[(f;:
4'1. IÍ g
lUT,l< k.
Lprrarr,ra
then
1T
cF
:
I
f l : LiF' f
is an r.couer-free family,
Pnoop. Trivial, choose
Lzuue 4.2. l./f(F)
I
=
FlF,eF
(";').
F, F' e g, T cF'].
Feg
withTtcFi
andT,,T,,...,r,eJ're)
and note
F(-F.U "'UF,.
Vol.51.
NO SET IS COVERED BY r OTHERS
1985
83
PRoop. In view of Lemma 4.LV(F) fulfills the following conditions:
/r(oC()' rr> lFI and (ii) Á'U...U A,lF tor A,,...,A, e /r(F).
(i)
Thus by Lemma
1
(Frankl I8)),
I
g
I
!
< (u, ') trolds.
Now Lemma 4.2 imphes that each F e
g
has at least
(k-t
rk)_
\tl \ I )):/k_r\
\t-tl
own subsets. Consequently,
_l\- /,r
rst(k
''\Í-|l'\;)
holds, yielding the desired upper bound.
5.
Proof of Theorem 2.3
Let 9s:{Fe 9:35 Cn lS l=r-1, such that SCF'e I implies F': F),
i.e. 9., denotes the family of members of .7 having an own subset of size smaller
than /. Clearlv. we have
ls,!=(,:,)
(3)
Lgltrun
5.1. IÍ F
e'
9 _ 9u and T,,T,,...,To*,E /r(F) then l U T l<
(d + I)t.
PRoop. Suppose for contradiction that lUfl =(d+I)t and let g{Tr,7.,...,T0*r, S,, Sr,..., S,-o-,} be a partition of F such that lS, I : t - 1. Then
for each Peg there exists a F"eT, F"lF with PCF. Hence Fc
U{R t P e q), a contradiction.
D
LsNaN,{a
5.2.
For F €
g _ g"
t/r(F) t= d(k,
(4)
(s)
(6)
we haue
_l),
-(u;o) iÍk>2t,d,
ift:2, k>'zd+z.
|./r(ol=(Í) _(;o)
l/r(F)l=(f)
Moreouer, equality holds
in
(5) or (6)
itr lUI].
e$:rtlf(F))l : k -
d.
P. PROOS ET AL.
Isr. J. Math.
Pnoop. Let us define m(k,t,d):max{lJ'f l:"t c(0, /f does not conrain
d * 1 pairwise disjoint members) where k > td, k, t, d are positive integers.
Erdös' Ko and Rado [6] proved that
/t
m(k,t,1): (';_-1\
i]
and
Íor k>2t
/k_1\
m(k.t.d)<d(\
;_i/
was shown by Frankl (cf. [7] or [9]). For k > k,(t, d) Erdös [3] proved that
/L\ (k_d\
m(k.t.o):(i/-r
t t.
Later
t :2,
ko(Í, d)<2t3d was established
m(k.2.
by Bollobás' Daykin and Erdös [2]. For
/ rl\
d): (;)+
d(k
- d)
was proved by Erdös and Gallai [5] (for k>(5dl2)+2). The uniqueness of the
optimal families was proved both in [2] and [5]. These results and Lemma 5.1
imply
(af(6).
!
From now on we suppose that one of the cases (a), (b), or (c) holds, i.e., (5) or
(6) is fulfilled. We apply the following theorem of Bollobás [i].
Lnuue 5.3. Let Ar,...,A^ and Br,...,B^ be finite
Ai n B, : A and A, n Bjl O holds for all il j. Then
\i
(7)
Moreouer,
'
iÍ |A,|:a,|B,|:b
lUÁ'l:IUB,|:a+b"
1
sets
and suppose that
=r
(lo',li,lt"'l):'
holds for all
i then equality holds in (7) only if
Divide 9F-9Fo into two parts:91:{tr'e g-go: l/f(F)l<(l)-(u;o)}, g,:
g - go- 9t. Then for each F e g2 we have a d-subset D(F)1F, such that
l(F-D(F)n F'l > t implies F : F'.
Now let Tr, Tr, . . . , T^ be the family of a// minimal own subsets of size at most
t of the members of 5, i.e., Tt C(F fl F') and F, F' e I imply F : F' , and for all
xe Ti there exists F'lF, F'e g such that (I-{x})CFnF'. Define
Vol.5l.l985
NO SET IS COVERED BY T OTHERS
":{
X-7,
X-7,-D(F)
85
cF e (9.u
Ti cF e 9,.
Tt
9,),
a Tjl AhoWs for all il j.If X, : X - Ti
then this fo|lows from the minimality of T,, i.e., TjÍ.T,. Suppose X:
X - T, - D(F). If 7 is a /-subset of ii U D(F), then either T: T, or
rnD@)laho|ds. Since I isanownsubsetof Some F,eg andF€9z,we
infer T,/ (i,1 u O1f;;, i.e., T1 o X,l A.
Clearly Xi n Ti
: A.Weclaim
that Xt
Now Lemma 5.3 yields
r>\
\
,-le"l
'' #, fro
l)X,,*tT,l\-l n \-(u;o)*',6.t(;o)
ln\ rt'ln-d1r":r'^
(,/
\ r )
'lT;,"";;"\ 'i,. 1 / \r-r/
Straightforward ca|culation shows that if n > 2dtt,|), then the coefficient oÍ | 9,|
is the smallest, hence we have
d\ (o
19,,'+!9,,+lg,l:,gl=('
.o\./
\ /. //I \ Í
if 9r:9,:O.Finally, to get the
extremal family we apply the second part of Lemma 5.3, which yields that each
D(F) is the same.
as desired. Moreover, equality can hold only
6.
Proof of Theorem 2.ó
We are going to use probabilistic methods.
Lpr'lr',ra 6.1. Let Y be an m-element set, m>2r. Then there exist (2r-1)uniform families 9r,...,9, such that PoP':A
for P,P'e?6 Igl=
(ml(2r-1))-I2r'\/m,lPnP'l<2 for all Pegi, P'e ?1 and s>m'21r'.
Pnoop. Let At, Az,. . . , A, be pairwise disjoint (2r - l)-element subsets of Y,
y : lm l(2r - 1)l . Consider 3s permutations chosen independently at random of
Y, 7r1,7r2,...,ÍÍz" where s : |m,/,lr,f. Define the family fr, as Irrt(Ai):I< j <
u). To obtain the families 9, we will delete the "bad" members of fr'.
For B € () we have that
Prob(B is covered by some members oÍ &,):
p,t(";')
(:\
\J/
4r'
1--1
m
.
P. ERDOS ET
86
AL
Isr. J. Math.
Hence we get
r(*
I)which
\ n. (\J,
(4\'.S
-- - I . (:\
\3/\tn'l '2m'
---r-
are covered by 9t, andft; as welt)
Finally we get
(8)
E(# R e UEl,: there exists
r
3st
R'eUEt,,lR
n R'i=
3)
8ro
=(Z /.2.3m<3s(8r'Vn).
Now. call a permutation zi "bad" If &i contains at least l2r'\/m members R
with the property lR nR'l=3 for some R'eU,+t{1,,. Then by (8) we have
E(# badq,,)<2s.
Thus there exists a choice of the random permutatioÍlS 7t, . . . , ÍÍ3,such that at
most 2s out of %tt,...,{l,* ZÍa bad. Suppose by symmetry {l,1,.'.,El" are not
bad. Each 9J; contains less than I2r'\/ mmembers R such that I R O R' l> 3 for
some R, . U,*,fri. Let 3t be the family obtained Írom gl, after deleting these
R. Then 9r,. . ., P. satisfy all the requirements.
n
Now the construction of the desired g C({), where X = 1I,2,..., n} is the
following. Let X: Y' U YrU. .. U Y" U Yo where
lY,
for all
s
0
l: ...:lY"l:^:
i<
i<
Y,, we get
7.
E
|
, a: Lr','lr'),
y,^ryi :CI
a. Take a copy of the families defined by Lemma ó.1 for each
: {P U lj} : P e E'j, I < i < j I m}. W e
. ., 3',. Finally, set
I
?\, gi,.
have
I
lr'n','1
=,2_(i
t
m
-
t)
(#-
rzr'
tm)
=
(;)
#-
o (n''").
Proof of Proposition 2.7
Let k and r be fixed. Let
g,(n,
family .7 such that for all FC g,
k) be the maximum size of an r-cover-free
TCF,lf l:t -1we
have an
F'lF, F'e g
with (F n F') l T.
Such a family .7 is called r-cover-free without small own subsets. Deleting
successively the members oÍ 9 having own (t _ l)-subsets we can always obtain a
g C g,
I
is without small own subsets. Obviously,
ts-El=(,lr)
Vol.51,
1985
NO SET IS COVERED BY r OTHERS
87
hence we have
Í,@, k)_
(, : 1)
=
g,1", k)< f,(n, k).
Hence it is sufficient to prove that for all e >0 and n there exists an AIs(n,e)
such that
(e)
s,(N.k)
holds whenever
I O'(o,', o, I (l))-'
N>No.
Let I C (f), I X l: , be an r-cover-free family without own parts of cardinality at most (r-1) such that |9|: g,(,,k).By Rödl's theorem (i.e. by (1)) for
1t' > AIo(n, e) there exists a (t, n, N)-packing I over the N-element set Y, with
/^r\ //n\
\,;)
t3t>(r_,)(';
)l
Replace each P e ? by a copy oÍ fi. We obtain an r-cover-free family on
points, yielding (9).
8.
N
Proof of Theorem 3.1
The upper bound of 3.1 comes from Proposition 2.1 using the obvious
fi@)=2oÍ,(n, k) and the Stirling formula.
The lower bound was obtained from Proposition 2.1, also, with k : nl4r.We
can get somewhat better lower bounds carrying out the proof given in [4] for the
case r =2.
9.
Proof of Theorem 3.3 and Proposition 3.4
Let fi
C2'
be an r-cover-free family and define
5,
: lF e I : F
has own subset of size most /).
Clearly, lg,l=(|).
Lpuue 9.1. If
Fe(g-9,)
l.I
I
and F1,Fr,...,Fte
u r,l> tg
i=t 'l
-
then
iy.
!
This lemma implies that:
(10) F,,...,F,*,e(g-g,) then Ii='+l
u ol=1r*
I
I
I)(tr+2)t2.
P. ERDOS ET AL.
For
r: e{nand t:
g,|í r, i.e.,
Ig _
Isr. J. Math.
l2le'1the right-hand side of
(10) is greater than n. Thus
/n\,
eti.< n|2|,,|
Ig,=(,;;^,-,|+
\ lLl t | /
The case r
:
Íor t>2.
1 follows from Proposition 3.4.
To prove Proposition 3.4 we apply induction on n. The statement is trivial,
e.g., for n < r. Suppose 9F c2*,lXl: ,, I is r-cover-free. If some F e F has a
1-element own subset, say {x }, then the statement follows by induction, applied
to 5 - IF]t, X - {x}. If 9,: A, and l9 l > r, then (10) implies
Ixl:r=(';'),
a contradiction. Thus
10. Final
l9
l< r < n holds.
remarks
The paper is a continuation of the earlier work of the authors [4] where they
dealt with the case r:2,i.e., A,ÉA,U Áz. The above topic is full of problems
which are related to designs and error-correcting codes.
OppN PRoarpNa. Suppose I c2',lXl= n,9 is r-cover-free, | 9l> n.For a
given r denote by n(r) the minimum of such n. Then by Proposition 3.4 we have
/
r+)\
\' ;' )=
n1r1<
r'+
o1r').
(The upper bound comes from the example of an affine plane of order at least
r + 1.) one can prove n (r) > (1 + o(I))2r,. We conjecture that |ím n(r)lr,: 1, or
even stronger n(r)> (r * 1)'. (We can prove this for r < 3.)
Added in proof. Theorem 3.1 was proved independently by Hwang and Sós
[12]. They apply the estimations oÍ f,(n) for group testing.
Rr,psnpNcss
1. B. Bollobás, on generalized graphs, Acta Math. Acad. Sci. Hungar. 16 (1965)' 44.7-452.
2. B. Bollobás' D. E. Daykin and P. Erdös, On the number of independent edges in a hypergraph,
Quart. J. Math. Oxford (2) 27 (1976),25-32.
3' P. Erdös, A problem of independent r-tuples, Ann' Univ. Budapest 8 (19ó5)' 93_95.
4. P. Erdös, P. Frankl and Z. Füredi, Families of finite sets in which no set is cotsered by the union
of two others, J. Comb. Theory, Ser. A 33 (1982)' 158_1óó.
Vol.51,
1985
NO SET IS COVERED BY r OTHERS
89
5. P. Erdös and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci.
Hungar. 10 (1959). 337-356.
6' P. Erdös, C' Ko and R. Rado, An intersection theorem for fnite seÍs, Quart. J' Math' oxford
(2) r2 (1961), 313-320.
7. P. Erdös and E. Szemerédi, Combinatorial properties oÍ a system of sets, J. Comb. Theory' Ser.
A 24 (1978),308-311.
8. P. Frankl, On Sperner families satisfying an additional condition, J. Comb. Theory Ser. A 20
(197ó)' 1-11.
9. P.Frankl, Ageneralintersectiontheotemforfinitesets,Ann.DiscreteMath.S(1980),43-49.
10. v. Rödl' On a packing and couering problem' Eur. J. Combinatorics 5 (1984).
11. J. Sperner, EinSatz iibet UntermengeneinerendlichenMenge,Matb.Z.2T (1928),544-548.
1,2. F.K. Hwang and V. T. Sós, Non-adaptioe hypergeometric g|oup testin& studia Sci. Math.
Hungar., to appear.