Adam Idzik

Gyula O.H. Katona

Rajiv Vohra

English

Research Papers in Economics

Journal of Combinatorial Theory, Series A 93, 281291 (2001)
Intersecting Balanced Families of Sets
Adam Idzik
Institute of Computer Science, Polish Academy of Sciences,
ul. Ordona 21, 01-237 Warsaw, Poland
E-mail: adidzikipipan.waw.pl
Gyula O. H. Katona
1
Alfre  d Re  nyi Institute of Mathematics, Hungarian Academy of Sciences,
P.O.B. 127, H-1364 Budapest, Hungary
E-mail: ohkatonarenyi.hu
and
Rajiv Vohra
Department of Economics, Brown University, Providence, Rhode Island 02912
E-mail: RajivVohrabrown.edu
Communicated by the Managing Editors
Received November 24, 1999
Suppose that any t members (t2) of a regular family on an n element set have
at least k common elements. It is proved that the largest member of the family has
at least k
1tn
1&1t elements. The same holds for balanced families, which is a
generalization of the regularity. The estimate is asymptotically sharp.  2001
Academic Press
Key Words: extremal problems; regular family; balanced family.
INTRODUCTION
The notion of a balanced family of sets has played an important role in
the analysis of the core of a cooperative game, a concept which is central
to cooperative game theory and economics. Bondareva [2, 3] and Shapley
[11] showed that a TU game possesses a non-empty core if and only if it
is a balanced game. A game is said to be balanced if a utility profile feasible
for every coalition belonging to a balanced family of coalitions is also
doi:10.1006jcta.2000.3078, available online at http:www.idealibrary.com on
281
0097-316501 35.00
Copyright  2001 by Academic Press
All rights of reproduction in any form reserved.
1 The work of the second author was supported by the Hungarian National Foundation for
Scientific Research Grant T029255 and the UNESCO Venice Office ROSTE, Grant 875.630.9.feasible for the grand coalition. Scarf [10] proved that every balanced
NTU game has a non-empty core. For further work in this area, see, for
example, Ichiishi [4], Ichiishi and Idzik [5], and Shapley [11]. Balanced
families of sets have also been useful in the study of other game theoretic
solution concepts, such as the bargaining set (see, for example, Maschler
[8] and Vohra [13]) and the kernel (Maschler et al. [9]).
In studying the non-emptiness of the bargaining set, Vohra [12, 13]
emphasized the importance of balanced families with the additional
property that all pairs of sets in the family have at least two common
elements. There, this property was used to establish that every NTU game
had a non-empty bargaining set if objecting coalitions were restricted in
size.
The present paper is concerned with a more systematic analysis of
balanced families which have intersecting sets. We show that if any t sets
(t2) of a balanced family have at least k common elements (k1), then
the largest set in this family has cardinality at least k
1tn
1&1t, where n is the
cardinality of the sum of all sets in this family. We also show that this
estimate is asymptotically sharp.
The first result along these lines was a theorem of Lova  sz [7] proving
that if a regular intersecting family (k=1) consists of sets of the same size
l, then l-n. Babai [1] proved the following generalization: if any two
sets (t=2) in a regular family have at least k elements in common, and all
sets in the family have the same size l, then l-kn. Finally Vohra [12]
observed that if any two sets in a balanced family have at least two
elements in common (t=k=2), then one of the members have size at
least -2n.
1. PRELIMINARIES
Let 2
[n] denote the family of all subsets of [n]=[1, ..., n] and R
n the
n-dimensional Euclidean space (R is the set of reals and R+ is the set of
nonnegative reals). By N we denote the set of natural numbers and by |S|
we denote the cardinality of a subset S/N. For any S #2
[n], let eS denote
the vector in R
n whose ith coordinate is 1 if i # S and 0 otherwise. We also
use the notation ei for e[i] and e for e[n].
Definition 1.1. A non-empty family B/2
[n] is balanced if there exist
[*S]S# B/R+, called balancing coefficients, such that S # B *SeS=e (or
equivalently, S #B : i # S *S=1 for every i #[n]).
Definition 1.2. A family D=[S1, ..., Sm]/2
[n] is d-regular (0
dm) if for every l #[n], |[j| l # Sj]|=d.
282 IDZIK, KATONA, AND VOHRAProposition 1.3. Every d-regular (1d) family is balanced.
Proof. Observe that 
m
i=1
1
d eSi=e. K
Definition 1.4. A family F=[S1, ..., Sm] is called t-wise k-intersecting
(m, t2, k1 are integers) if it satisfies the condition
|Si1 & }}}&Sit|k for any t members of F. (1.1)
2. LOWER ESTIMATE ON THE LARGEST MEMBER
Theorem 2.1. Let m, t2, k1 be integers. Assume that the balanced
family B=[S1, ..., Sm]/2
[n] is t-wise k-intersecting. Then B contains a
member Sj such that
|Sj |k
1tn
1&1t. (2.1)
Proof.L e t*j be the balancing coefficient of the the set Sj . Introduce the
notation s=max[|Sj |:Sj # B].
For i #[n], let
S(i)=[Sj # B | i # Sj],
di=|S(i)|,
*(i)= :
[j| Sj # S(i)]
*j .
Consider the matrix M whose j-th column is *j eSj . Adding all the entries
in the ith row yields *(i)=1. Therefore the sum of all entries of M is n.
First adding the columns we obtain
:
m
j=1
*j |Sj |=n.
Clearly, s|Sj | for all j, hence we have
s :
m
j=1
*jn
or
s
n

m
j=1 *j
. (2.2)
283 BALANCED FAMILIES OF SETSNow, consider all possible products of t entries in the same row of M,
where repetition is allowed and the order of the entries does matter. Of
course, only those products are interesting which contain only non-zero
factors. Denote the sum of all these products by A.
Determine the sum of the products in one row. We may suppose without
loss of generality that the non-zero entries in this row are *1, ..., *r(rm),
where 
r
i=1 *i=1. Our sum in question is
:
( j1, ..., jt), 1 jir
*j1 }}}*jt=(*1+}}}+ *r)
t=1.
Therefore A=n.
On the other hand, classify the terms occuring in A according to the
sequences (u1, ..., ut), 1uim. These classes are obviously disjoint. They
contain identical terms: *u1 }}}*ut. Let us see that the number of these terms
is at least k. Since |Su1 & }}}&Sut|k holds, the matrix M contains at
least k rows where all t columns u1, ..., ut are non-zero. Therefore
Ak :
(u1, ..., ut), 1uim
*u1 }}}*ut=k(*1+}}}+*m)
t.
This proves the inequality
nk(*1+}}}+ *m)
t. (2.3)
Substituting (2.3) in (2.2) we have
sk
1tn
1&1t. K
By Proposition 1.3 we obtain the following special case.
Corollary 2.2. Let m, t2, k1 be integers. Assume that the
d-regular (1d) family D=[S1, ..., Sm]/2
[n] is t-wise k-intersecting. Then
D contains a member Sj such that
|Sj |k
1tn
1&1t. K
If t=2 and |Si|=|Sj | for all i, j #[n], Corollary 2.2 reduces to Lemma
3 of Babai [1].
Now we give an alternative proof for the regular case. This actually gives
an upper bound on the number of sets, too.
Theorem 2.3. Let m, t2, k1 be integers. Assume that the family
B=[S1, ..., Sm]/2
[n] is t-wise k-intersecting. Let di=|[j| i # Sj]| for
i#[n]. Then
m\

n
i=1 d
t
i
k +
1t
(2.4)
284 IDZIK, KATONA, AND VOHRAand B contains a member Sj such that
|Sj |k
1t 
n
i=1 di
(
n
i=1 d
t
i)
1t . (2.5)
Proof. Consider the 0,1 matrix M$ whose jth column is eSj . The
number of 1's in the ith row is di. The total number of non-zero products
of t entries in the same row of M$, where repetition is allowed and the
order of the entries does matter is
:
n
i=1
d
t
i .
By the intersecting condition of the family, this is at least km
t. This
inequality proves (2.4).
The total number of 1's in M$i s
n
i=1 di , therefore there is a column
containing at least 
n
i=1 dim 1's. The substitution of (2.4) proves (2.5). K
If d1=}}}= dn=d that is the family is d-regular then (2.4) gives the
upper bound
md\
n
k+
1t
.
On the other hand, (2.5) leads to Corollary 2.2.
3. CONSTRUCTIONS
Denote the minimum size of the largest member of a balanced t-wise
k-intersecting family on n elements by f(n, t, k). Theorem 2.1 states that
k
1tn
1&1t f(n, t, k).
In the present section we give an upper estimate on f(n, t, k).
Let q be a prime power, t2 and PGt(q)at-dimensional finite projec-
tive geometry over the q-element finite field. The number of points in
PGt(q)i sq
t+}}}+q+1 (see, e.g. [6]). The size of a t&1-dimensional
hyperplane is q
t&1+}}}+ q+1. The intersection of t such distinct
hyperplanes is always one point. Therefore we have a family of q
t&1+
}}}+q+1-element subsets of the q
t+}}}+q+1-element set so that the
intersection of any t distinct ones has 1 element.
Replace each element in the above example by a k-element set. Then a
family of k(q
t&1+}}}+q+1)-element subsets of a k(q
t+}}}+ q+1)
285 BALANCED FAMILIES OF SETSelement set is obtained so that the intersection of t distinct subsets has
exactly k elements. This proves
f(k(q
t+}}}+ q+1), t, k)k(q
t&1+}}}+q+1). (3.1)
Using this idea, one can easily prove that the estimate of Theorem 2.1 is
asymptotically correct.
Theorem 3.1. f(n, t, k)tk
1tn
1&1t
holds for fixed t and k when n tends to infinity.
To prove Theorem 3.1 we need the following two lemmas.
Lemma 3.2. For any 0<=<1, positive k and t and sufficiently large n
there is a prime number q satisfying
(1&=) n<k(q
t+}}}+q+1)n. (3.2)
Proof. Let us start with the well-known fact that, for fixed 0<:<;
and sufficiently large n there is a prime number q such that :n<q<;n.
Apply this statement for the necessary constants and n
1t:i fn is large
enough, there is a prime number q such that
(1&=)
1t n
1t
k
1t <q<
(1&=2)
1t n
1t
k
1t .
This implies (3.2) for large n. K
Lemma 3.3. Given the positive integers da and u, one can find subsets
B1, ..., Ba of the u-element set U so that every element of U is contained in
exactly d subsets, and |Bi|W
du
a X (1ia). The subsets Bi are not
necessary different.
Proof. Define b and r by du by a: du=ab+r,0  r<a. It is easy to see
that one can find sets with sizes |B1|=}}}=|Ba&r|=b,| Ba&r+1|=
}}}=|Ba|=b+1 in such a way that the elements of U are contained in the
same number of sets. The sum of the degrees is equal to the sum of the
sizes: (a&r) b+r(b+1)=ab+r=du. Hence it follows that the degrees are
all d. K
Proof of Theorem 3.1. Choose q according to Lemma 3.2. Follow the
above geometric construction with a slight modification. Replace each
point of PGt(q)b yak-element set. The new underlying set K has
k(q
t+}}}+ q+1) elements. The t&1-dimensional hyperplanes of PGt(q)
286 IDZIK, KATONA, AND VOHRAare enlarged, too, each of these enlarged sets has k(q
t&1+}}}+ q+1)
elements. Denote them by Hi (1iq
t+}}}+ q+1). This family on K
is t-wise k-intersecting and regular of degree q
t&1+}}}+ q+1. Let
0u=n&k(q
t+}}}+ q+1) and take a u-element set U disjointly to K.
Apply Lemma 3.3 with a=q
t+}}}+ q+1, d=q
t&1+}}}+q+1 and u.
Using (3.2) we obtain that the sizes of the B's are at most

du
a|

(q
t&1+}}}+ q+1) =n
q
t+}}}q+1 |

k=
1&=
(q
t&1+}}}+ q+1). (3.3)
The family Hi _Bi (1iq
t+}}}+ q+1) on the n-element underlying
set K_U is obviously t-wise k-intersecting and regular (of degree
q
t&1+}}}+ q+1). The sizes can be upperbounded using (3.2) and (3.3),
|Hi _Bi|\
1+
=
1&=+
k(q
t&1+}}}+ q+1)
\
1+
=
1&=+
k(q+1)
t&1\
1+
=
1&=+
k
1t(k(q+1)
t)
(t&1)t.
Here k(q+1)
t<k(1+=)(q
t+}}}+ q+1)(1+=) n holds for large n's.
Therefore
|Hi _Bi|<\
1+
=
1&=+
k
1t(1+=)
(t&1)tn
(t&1)t.
We have proved
f(n, t, k)\
1+
=
1&=+
(1+=)
t&1t k
1tn
(t&1)t
for sufficiently large n's. This inequality, combined with Theorem 2.1,
finishes the proof. K
The case t=2, k=1,n=q
2+q+1 is much easier.
Theorem 3.4. f(q
2+q+ 1 ,2 ,1 ) = q+1.
Proof. Theorem 2.1 gives f(q
2+q+1, 2, 1)W-q
2+q+1X. The
obvious q
2<q
2+q+1<(q+1)
2 proves W-q
2+q+1X=q+1. (3.1) com-
pletes the proof. K
Theorem 3.4 was proved by Lova  sz [7] in the special case when the
sizes of the sets in the family are all equal, i.e., for uniform d-regular
families.
287 BALANCED FAMILIES OF SETS4. FURTHER QUESTIONS
The degree of the regular family in our construction is relatively large, its
order of magnitude is dtk
1t&1n
1&1t. The obvious question arises whether
the size of the largest set in the family increases when the degree (of
regularity) is fixed to be small. Let f(n, t, k, d) be the minimum (for
families) of the size of the largest member in a t-wise k-intersecting regular
family of degree d on an n-element underlying set.
The above definition makes sense at all when td. This is why studying
f(n, t, k, t) is a good starting point.
Theorem 4.1. f(k (
l+t&1
t ), t, k, t)=k (
l+t&2
t&1 ) holds for all integers t2,
k1, l1.
Proof. Let us start with the construction for the case k=1. The under-
lying set of the family will consist of all t-element subsets of [l+t&1], that
is, the t-element subsets will play the role of elements. Let the family
D(t, l)=[S1, ..., Sm] be defined by Si=[A | i # A,| A|=l+t&2](1im
=l+t&1). It is easy to see that Si1 & }}}&Sit=[[i1, ..., it]], i.e.,
|Si1 & }}}&Sit|=1 holds for distinct i1, ..., it. [i1, ..., it] is an element of Si
iff i=i1, ..., it, that is, the degree of each element of the underlying set is t.
Finally, |Si|=(
l+t&2
t&1 ) proves
f\\
l+t&1
t +
, t,1 ,t+
\
l+t&2
t&1 +
.
If the elements of the underlying set are replaced by finite sets, the new
family will be t-wise intersecting of degree t, again, only the sizes of the
members will become larger. Denote the class of these modified families by
2(t, l). Given a family F on [n] we say that two elements 1i, jn are
equivalent if i # F and j # F hold for the same members F# F. The set of
equivalent elements is called an atom. It is easy to see that each atom of
D(t, l) has one element. On the other hand the elements obtained from one
element when making another family D #2(t, l) from D(t, l) form an atom.
The family D # 2(t, l) with atoms of size k serves as a construction
proving
f\
k\
l+t&1
t +
, t, k, t+
k\
l+t&2
t&1 +
.
Before proving the other direction we will show a lemma stating
that these are the only constructions. Let k1, t2, l1 be integers. The
288 IDZIK, KATONA, AND VOHRAset of families D consisting of l+t&1 sets, being t-wise k-intersecting
and regular of degree t is denoted by 2*(t, l, k), moreover 

k=1
2*(t, l, k)=2*(t, l).
Lemma 4.2. The elements of 2*(t, l) differ only in the sizes of the atoms
and the permutation of the elements.
Proof. Use induction on t and l.I fl=1, t is arbitrary then the degree
condition ensures that the family D # 2*(t,1 ,k) consists of t identical sets.
The statement is trivial for this case.
Let t=2,l>1 and suppose that the lemma is true for t=2, l&1. Take
a family D # 2*(2, l, k). Its members are denoted by D1, ..., Dl+1. The
intersections D1 &Di (2il+1) must form a partition of D1. Here
|D1 &Di|k. On the other hand Di&D1 (2il+1) is a family of l sets,
it is 2-wise k-intersecting and regular of degree 2, that is, it is in
2*(2, l&1). By the induction hypothesis it is uniquely determined in the
given sense. The statement is proved for this case.
Suppose that t>2, l>1 and the statement is true for t&1 with all values
of l and for t with l&1. Choose a family D # 2*(t, l, k). The t&1-wise
intersections of D1 &Di (2il+t&1) form a partition of D1 with
classes of size at least k. Therefore the family D1 &Di (2il+t&1) is
in 2*(t&1, l&1,k), by the induction, it is uniquely determined (in the
given sense). On the other hand Di&D1 (2il+t&1) is a family of
l+t&2 sets, it is t-wise k-intersecting and regular of degree t, that is, it is
in 2*(t, l&1, k). By the induction hypothesis it is uniquely determined in
the given sense. K
Corollary 4.3. 2*(t, l)=2(t, l).
Proof. It is easy to check that D(t, l)#2*(t, l). Lemma 4.2 completes
the proof. K
Now return to the proof of Theorem 4.1. By the corollary above, it is
enough to consider the members of 2(t, l). So, suppose that D # 2(t, l) has
atoms of size kai where
k\
l+t&1
t +
n (4.1)
and
: ai=n (4.2)
289 BALANCED FAMILIES OF SETShold. Denote the members of D by T1, ..., Tl+t&1. The regularity of D and
(4.2) imply
:
l+t&1
i=1
|Ti|=: aj t=nt.
Hence there exists an i satisfying

nt
l+t&1|
|Ti|. (4.3)
On the other hand, if the sizes of the atoms are chosen nearly equal, then
this inequality is sharp. The result is a monotone function of l, therefore we
have to choose l the largest possible, satisfying (4.1). By this, the value of
f(n, t, k, t) is determined for all values of n. In the special case of the
theorem when n=k(
l+t&1
t ) the left hand side of (4.3) is really k(
l+t&2
t&1 ). K
It is easy to deduce the following result from Theorem 4.1.
Corollary 4.4. For fixed t and k and sufficiently large n
f(n, t, k, t)t
t
(t!)
1t k
1tn
1&1t
holds.
It is surprising that the order of magnitude of the largest set is almost the
same under this strong condition on the degree as in the case of unlimited
degree (Theorem 3.1).
ACKNOWLEDGMENT
We are grateful to Stanis*aw Bylka, Zolta  n Fu  redi, and Imre Z. Ruzsa for valuable
discussions.
Note added in proof. (1) Noga Alon remarked that a family is balanced iff (non-
negative integral) multiplicities can be associated with the members in such a way that the so
obtained ``multi-family'' is regular. (It is easy to see that the balancing coefficients in Defini-
tion 1.1 can be chosen rational. Multiply the equation with the lowest common multiple d of
the denominators of these rational numbers. We obtain S # B lSeS=(d, ..., d).) The proof of
Theorem 2.3 is valid for multi-families, therefore it gives an alternative proof also for the
general case of balanced families.
(2) ?-balanced families are generalizations of balanced families (see, e.g., [5, 11] for the
definition. Unfortunatelly, our theorems are no longer true for this class of families as the
``star'' consisting of two-element sets shows.
290 IDZIK, KATONA, AND VOHRAREFERENCES
1. L. Babai, Long cycles in vertex transitive graphs, J. Graph Theory 3 (1979), 301304.
2. O. N. Bondareva, Teoriia iadra v igre n lits, Vestnik Leningrad Univ. Ser. Math. 17(13),
No. 4 (1962), 141142.
3. O. N. Bondareva, Nekotorye primeneniia metodov linejnogo programmirovniia k teorii
kooperativenykh igr, Problemy Kibernet. 10 (1963), 119139.
4. T. Ichiishi, ``Game Theory for Economic Analysis,'' Academic Press, New York, 1983.
5. T. Ichiishi and A. Idzik, Theorems on closed coverings of a simplex and their applications
to cooperative game theory, J. Math. Anal. Appl. 146 (1990), 259270.
6. J. H. van Lint and R. M. Wilson, ``A Course in Combinatorics,'' Cambridge Univ. Press,
Cambridge, UK, 1992.
7. L. Lova  sz, On minimax theorems of combinatorics, Mat. Lapok 26 (1975), 209264.
[In Hungarian]
8. M. Maschler, The inequalities that determine the bargaining set M
(i)
1 , Israel J. Math. 4
(1966), 127134.
9. M. Maschler, B. Peleg, and L. S. Shapley, The kernel and bargaining set for convex
games, Internat. J. Game Theory 1 (1972), 7393.
10. H. Scarf, The core of an N person game, Econometrica 35 (1967), 5069.
11. L. S. Shapley, On balanced sets and cores, Naval Res. Logist. Quart. 14 (1967), 453460.
12. R. Vohra, On a bargaining set with restricted coalition size, working paper, Department
of Economics, Brown University, 1987.
13. R. Vohra, An existence theorem for a bargaining set, J. Math. Econom. 20 (1991), 1934.
291 BALANCED FAMILIES OF SETS

Civil Conflict: Ended Or Never Ending?

http://www.econ.brown.edu/~rvohra/papers/wp99-33.pdf

Civil Conflict: Ended Or Never Ending?

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics