AbstractThe subsets A,B of the n-element X are said to be s-strongly separating if the two sets divide X into four sets of size at least s. The maximum number h(n,s) of pairwise s-strongly separating subsets was asymptotically determined by Frankl (Ars Combin. 1 (1976) 53) for fixed s and large n. A new proof is given. Also, estimates for h(n,cn) are found where c is a small constant

Katona, Gyula

Katona, Gyula O.H.

Elsevier - Publisher Connector 

Discrete Applied Mathematics 137 (2004) 87–95
www.elsevier.com/locate/dam
Strong qualitative independence
Gyula O.H. Katona
Alfred Renyi Institute of Mathematics, Hungarian Academy of Science, P.O.B. 127, Budapest H-1364,
Hungary
Received 20 October 2001; received in revised form 12 August 2002; accepted 15 October 2002
Abstract
The subsets A; B of the n-element X are said to be s-strongly separating if the two sets divide
X into four sets of size at least s. The maximum number h(n; s) of pairwise s-strongly separating
subsets was asymptotically determined by Frankl (Ars Combin. 1 (1976) 53) for 7xed s and
large n. A new proof is given. Also, estimates for h(n; cn) are found where c is a small constant.
? 2003 Elsevier B.V. All rights reserved.
MSC: primary 90B40; secondary 05D05; 94A99; 94B99
Keywords: Search; Qualitatively independent sets; Hadamard matrix; Code
1. Introduction
Let X be a 7nite set of n elements. The following notion was introduced by Mar-
czewski [17]. The subsets A; B ⊂ X are called qualitatively independent if they divide
X into four non-empty parts that is A∩B; A∩ @B; @A∩B; @A∩ @B are all non-empty. The
signi7cance of this notion in search theory lies in the consequence that after knowing if
an unknown x is in A or not we cannot decide the same question for B, independently
on the answer for the 7rst question. The family F ⊂ 2X is called independent if their
members are pairwise qualitatively independent.
RAenyi asked [23] the question what is the maximum size of an independent family
in an n-element set. The answer was found by the present author ([12], see also [23]),
and independently by Brace and Daykin [5], BollobAas [4] and Kleitman and Spencer
[13] (see also [24]).
E-mail address: ohkatona@tenyi.hu (G.O.H. Katona).
0166-218X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0166-218X(03)00190-2
88 G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95
Theorem 1.1. The maximum number of pairwise qualitatively independent sets in an
n-element set is(
n− 1
 n2
)
:
If the family F is independent then A; B∈F; A = B implies A ⊂ B that is F
is inclusion free. The maximum size of an inclusion-free family is determined by a
classical theorem of Sperner [25].
Theorem 1.2. If F is an inclusion-free family in an n-element set then
|F|6
(
n
	 n2

)
:
The following sharpening of Sperner’s theorem is also well known as the YBLM-
inequality (earlier LYM) (see [3,16,18,27]). It involves the sizes of the “levels” of the
family. If F is a family, de7ne fi(F) = |{F : F ∈F; |F |= i}|.
Theorem 1.3. If F is an inclusion-free family in an n-element set then
n∑
i=0
fi(F)(
n
i
) 6 1
holds.
Kleitman and Spencer [13] introduced the notion of k-indpendence. A family F ⊂
2X is called k-independent iL
k⋂
i=1
Aii = ∅
holds for any choice of distinct A1; : : : ; Ak ∈F and i = 0; 1 where A0 = A; A1 = @A that
is when any k members divide X into 2k non-empty parts. Let f(n; k) denote the
maximum size of a k-independent family. They proved the following theorem.
Theorem 1.4 (Kleitman and Spencer [13]).
2c12
−k k−1n6f(n; k)6 2c22
−k n:
See also [1].
A set A can be considered as a partition (A; @A). A generalization in this direction is to
consider partitions into r parts that is r-partitions. Two r-partitions are called qualita-
tive independent if all the r2 intersections of the classes are non-empty. The maximum
number of pairwise qualitatively independent r-partitions is denoted by g(n; r). As this
G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95 89
is exponential in n and an exact formula for g(n; r) is hopeless, it is suMcient to
consider the exponent:
qr = lim sup
n→∞
1
n
log g(n; r):
After some preliminary results [19–22,15] Gargano et al. [9] have determined the exact
exponent.
Theorem 1.5 (Gargano et al. [9]).
qr =
2
r
(26 r):
For a recent interesting result concerning qualitative independence see [14].
A family S ⊆ 2X is s-strongly separating iL all four intersections A∩B; A∩ @B; @A∩
B; @A ∩ @B are of size at least s for any two distinct members A; B∈S. The maximum
size of an s-strongly separating family is denoted by h(n; s). The determination of
h(n; s) was asked in [12]. It has been asymptotically answered by Frankl for 7xed s.
Theorem 1.6 (Frankl [8]).
d1(s)
2n
ns−1=2
6 h(n; s)6d2(s)
2n
ns−1=2
:
where
d1(s) =
√
2

1
2s
− 
and
d2(s) =
√
2

2s−2(s− 1)! + :
The author returned to this question now because Benjamin Weiss [26] suggested
the following related problem. Find the maximum number of sequences of length n
over the alphabet {0; 1; 2} such that if a = (a1; : : : ; an); b = (b1; : : : ; bn) are two such
sequences then every pair (c; d) c; d∈{0; 1; 2} occurs either 0 times or at least twice
among the pairs (ai; bi) (16 i6 n). The analogous problem for {0; 1} is just h(n; 2).
However we only found a new proof for Frankl’s theorem and some estimates for
the case when s= cn where c is a small constant.
In the new proof of the upper estimate (in Theorem 1.6) it is suMcient to use the
following weaker condition. A family F is called s-di:bounded iL |A− B|¿ s holds
for any two members of F. Let us mention that a family is 1-diLbounded iL A ⊂ B for
any two distinct members, in other words, iL the family is inclusion free. It is obvious
that if S is s-strongly separating then it is s-di:bounded. The following theorem is a
generalization of Theorem 1.3.
90 G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95
Theorem 1.7. Let s be a positive integer. If F is an s-di:bounded family in an
n-element set then
n∑
i=s−1
fi(F)
(
i
s− 1
)
(
n
i − s+ 1
)6 1: (1.1)
Bey [2] called the author’s attention to the fact that Theorem 1.7 is a special case
of Theorem 4 of [28] (i = t in its notation).
Corollary 1.8. Let s be a positive integer. If F is an s-di:bounded family in an
n-element set then
|F|6d3(s) 2
n
ns−1=2
where
d3(s) =
(√
2

2s−1(s− 1)! + 
)
:
If S is s-strongly separating then the family
F= {S; @S: S ∈S}
is s-diLbounded. Here |S|6 12 |F| holds and this yields the following improvement of
Corollary 1.8.
Corollary 1.9. Let s be a positive integer. If S is an s-strongly separating family in
an n-element set then
|S|6
(√
2

2s−2(s− 1)! + 
)
2n
ns−1=2
:
In Section 3 the case when s=cn is considered. We have a non-trivial upper estimate
only when c is small.
Theorem 1.10.
h(n; cn)6 2n(−(1=2) log c−1:099)+o(n)
where c6 0:099.
2. The case of !xed s
Proof of Theorem 1.7. De7ne the r-shadow of a family F in the following way.
r(F) = {A− B: B ⊂ A∈F; |B|= r}:
G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95 91
Let A; B∈F be distinct members and choose an s− 1-element subset C of A. Here
|A−B|¿ s implies A−C * B. If D is an s−1-element subset of B then A−C * B−D
follows. This is obviously true if A=B but C = D. Therefore s−1(F) is inclusion free.
Determine the sizes of the levels of s−1(F). The sets on level j = i − s + 1
(s− 16 i6 n) are obtained from sets on level i in F:
fj(s−1(F)) = fi(F)
(
i
s− 1
)
:
The YBLM-inequality for the family s−1(F) is exactly (1.1).
Proof of Corollary 1.8. Theorem 1.7 will be used. Find the minimum of the ratio(
i
s− 1
)
(
n
i − s+ 1
)
in the terms of (1.1). The above ratio is equal to(
n+ s− 1
s− 1
)
(
n+ s− 1
i
) :
Its minimum is(
n+ s− 1
s− 1
)
(
n+ s− 1
	 n+s−12 

) :
Therefore (1.1) implies
|F|=
n∑
i=s−1
fi(F)6
(
n+ s− 1
	 n+s−12 

)
(
n+ s− 1
s− 1
) :
The application of the Stirling formula leads to the desired result.
It is easy to see that
d3(s) =
√
2

2s−1(s− 1)! + 
for large enough n.
Now Corollary 1.9 easily follows.
92 G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95
In the proof of the lower estimate of Theorem 1.6 the following result will be
used from coding theory. A codeword is a 0,1 sequence of length n. We say that
the codeword has weight w if the number of 1s is w. The Hamming distance of two
codewords is the number of diLerent digits. Let A(n; 2; w) be the maximum number
of codewords of weight w with pairwise Hamming distance at least 2.
Theorem 2.1 (Graham and Sloane [10]). Let q¿ n be a prime power. Then(
n
w
)
q−1
6A(n; 2; w):
Proof of Theorem 1.6. The upper estimate is a consequence of Corollary 1.9.
The construction of the lower estimate is based on Theorem 2.1. Use the theorem for
n− s; 2s; 	 n2
− s. The codewords de7ne, in an obvious way, subsets of an n− s-element
set Y with pairwise diLerences at least s. Denote this family by F′. Add a disjoint
set U to Y , where |U |= s. The extended underlying set is X = Y ∪ U . De7ne
F= {F ∪ U : F ∈F′};
that is U is contained in all the sets. Since the diLerence of any two members of F′
have size at least s, the same holds for any two members of F, that is, F satis7es
the conditions of the theorem. By Theorem 2.1 we have(
n− s
	 n2
 − s
)
qs−1
6 |F|: (2.1)
By the prime number theorem there is a prime q satisfying n6 q6 n+o(n). This fact
(2.1) and the Stirling formula proves the lower estimate with
d1(s) =
√
2

1
2s
− 
for large enough n.
3. The case when s = cn
The other end: In the previous section, the case when s is much smaller than n was
considered. Suppose now that n is divisible by 4 and s=n=4. Let S be an n=4-strongly
separating family. Then A; B∈S; (A = B) divide X into four parts of n=4 elements
each. Associate a vector with coordinates 1;−1 with a member A of S writing 1 in
the ith position iL the ith element of X is in A. Denote the vectors obtained in this
way from the members of S by v1; v2; : : : vm. It is easy to see that the inner product
vivj is 0 for 16 i¡ j6m: Let v0 have 1 in each coordinate. Then v0vi =0 also holds
(16 i6m). That is, v0; v1; : : : ; vm are pairwise orthogonal vectors in an n-dimensional
space. Then m6 n− 1 follows. The following little statement is obtained.
G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95 93
Proposition 3.1. Let n divisible by 4. Then
h
(
n;
n
4
)
6 n− 1
with equality i: there is a Hadamard matrix.
Consider now the case when n=4s+ t with a small positive integer t. Then |vivj|6 t
must hold for all 06 i¡ j6m. Since |vi|=n (06 i6m) therefore the normed version
of vi is ui=vi=|vi|=vi=
√
n. Their lengths are 1, on the other hand |uiuj|6 t=n holds for
06 i¡ j6m. We arrived to a geometric problem: what is the maximum number m of
points on the surface of the n-dimensional unit ball Bn (it is called the n−1-dimensional
spherical space Sn−1) if the angle ’ between any two vectors determined by these
points satis7es |cos’|6 t=n.
A slightly diLerent version of this problem is widely studied: what is the maximum
number M (d; ’) of points on the surface of the d=(n−1)-dimensional spherical space
Sd if the angle between any two vectors determined by these points is at least ’. For
good surveys see [7,6].
In our case the vectors u0; u1; : : : ; um;−u0;−u1; : : : ;−um possess the property that
their pairwise angle is at most arccos t=(d + 1). Denote the maximum number of
points P1; : : : ; Pm in the d-dimensional spherical space such that the angle between
any two of these points and their opposites is at least ’ by N (d; ’). The inequal-
ity 2N (d; ’)6M (d; ’) is obvious. It does not always hold with equality as the case
d=2; ’= arccos 13 shows, since M (2; arccos
1
3 ) = 9 while N (2; arccos
1
3 ) = 4. We have
proved the following lemma.
Lemma 3.2.
h(4s+ t; s)6N
(
4s+ t − 1; arccos t
4s+ t
)
− 1
6
1
2
M
(
4s+ t − 1; arccos t
4s+ t
)
− 1:
In order to obtain good estimates on h(4s+t; s) for 7xed t and large s one should study
the corresponding values of M (d; ’) and N (d; ’). E.g. it seems to be not diMcult to
prove the inequality h(4s+1; s)6 4s+1, but h(5; 1)=4 by Theorem 1.1 showing that
the equality does not always hold.
The middle case: Here we want to 7nd h(n; cn) where c is a constant and n is large.
Our only result is Theorem 1.10.
Proof of Theorem 1.10. This upper estimate is a trivial combination of Lemma 3.2
and a theorem of KabatjanskiTU and LevenTstein [11] claiming
M (d; ’)6 (1− cos’)−d=22−0:099d++o(d) (’6 62:9974 : : :◦):
We have no reasonable lower estimate. Theorem 2.1 does not help, since its estimate
tends to 0 when s= cn and n tends to ∞.
94 G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95
4. Open problems
Restriction with the pairwise entropy: Let us repeat a problem posed in [12]. If
p= (p1; : : : ; pm) is a probability distribution then its entropy is
H (p) =
m∑
i=1
− pi logpi:
De7ne the entropy of a pair of sets A; B ⊂ X as the entropy of the probability distri-
bution( |A ∩ B|
|X | ;
| @A ∩ B|
|X | ;
|A ∩ @B|
|X | ;
| @A ∩ @B|
|X |
)
:
What is the maximum number of subsets A1; : : : ; Am of an n-element set X if
the entropy of any pair Ai; Aj (i = j) is at least ,? The case , = 4 is solved by
Proposition 3.1.
Combining the problems: The combinations of the problems of Theorems 1.4, 1.5
and 1.6 give rise to many new problems.
Acknowledgements
The work was supported by the Hungarian National Foundation for Scienti7c Re-
search grant numbers T029255 and T037846.
References
[1] Alon, Noga, Explicit construction of exponential sized families of k-independent sets, Discrete Math.
58 (1986) 191–193.
[2] C. Bey, personal communication.
[3] B. BollobAas, On generalized graphs, Acta Math. Hungar. 16 (1965) 447–452.
[4] B. BollobAas, Sperner systems consisting of pairs of complementary subsets, J. Combin. Theory Ser. A
15 (1973) 363–366.
[5] A. Brace, D.E. Daykin, Sperner-type theorems for 7nite sets, in: D.J.A. Welsch, D.R. Woodall
(Eds.), Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford), Inst. Math. Appl.,
Southend-on-Sea, 1972, pp. 18–37.
[6] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, 2nd Edition, Springer, New York,
1993.
[7] G. Fejes TAoth, Spherical space, in: J.E. Goodman, J. O’Rourke (Eds.), Handbook of Discrete and
Computational Geometry, CRC Press, New York (1997) pp. 32–34.
[8] P. Frankl, An extremal problem of coding type, Ars Combin. 1 (1976) 53–55.
[9] L. Gargano, J. KYorner, U. Vaccaro, Sperner capacities, Graphs and Combin. 9 (1993) 31–56.
[10] R.L. Graham, N.J.A. Sloane, Lower bounds for Constant Weight Codes, IEEE Trans. Inform. Theory
26 (1980) 37–43.
[11] G.A. KabatjanskiTi, V.I. LevenTstein, Bounds for packings on a sphere and in space, Problemy Peredachy
Inf. 14 (1) (1978) 3–25;
G.A. KabatjanskiTi, V.I. LevenTstein, Bounds for packings on a sphere and in space, Problems Inf. Trans.
14 (1) (1978) 1–17.
[12] G.O.H. Katona, Two applications (for search theory and truth functions) of Spener type theorems,
Periodica Math. Hungar. 3 (1973) 19–26.
G.O.H. Katona /Discrete Applied Mathematics 137 (2004) 87–95 95
[13] D. Kleitman, J. Spencer, Families of k-independent sets, Discrete Math. 6 (1973) 255–262.
[14] J. KYorner, A. Monti, Delta-systems and qualitative (in)dependence, J. Combin. Theory 99 (2002)
75–84.
[15] J. KYorner, G. Simonyi, A Sperner-type theorem and qualitative independence, J. Combin. Theory Ser.
A 59 (1992) 90–103.
[16] D. Lubell, A short proof of Sperner’s lemma, J. Combin. Theory 1 (1966) 299.
[17] E. Marczewski, IndAependance d’ensembles et prolongement de mesures, Colloq. Math. 1 (1948)
122–132.
[18] L.D. MeTsalkin, A generalization of Sperner’s theorem on the number of subsets of a 7nite set, Teor.
Verojatnost. i Primen. 8 (1963) 219–220 (in Russian with German summary).
[19] S. Poljak, A. Pultr, V. RYodl, On qualitatively independent partitions and related problems, Discrete
Appl. Math. 6 (1983) 193–205.
[20] S. Poljak, A. Pultr, V. RYodl, On the dimension of Kneser graphs, in: L. LovAasz, V.T. SAos (Eds.),
Algebraic Methods in Graph Theory, Szeged, Hungary (1978) (Coll. Soc. Math. J. Bolyai) 25, North
Holland, Amsterdam (1981), pp. 631–646.
[21] S. Poljak, V. RYodl, Orthogonal partitions and covering of graphs, Czech Math. J. 30 (1980) 475–485.
[22] S. Poljak, Z. Tuza, On the maximum number of qualitatively independent partitions, J. Combin. Theory
Ser. A 51 (1989) 111–116.
[23] A. RAenyi, Foundations of Probability, Wiley, New York, 1971.
[24] J. SchYonheim, On a problem of Purdy related to Sperner systems, Canad. Math. Bull. 17 (1974)
135–136.
[25] E. Sperner, Ein Satz Yuber Untermengen einer endlichen Menge, Math. Z. 27 (1928) 544–548.
[26] Weiss, Benjamin, personal communication.
[27] K. Yamamoto, Logarithmic order of free distributive lattices, J. Math. Soc. Japan 6 (1954) 347–357.
[28] Zhang, Zhen, LYM-TYpe Inequalities for tEC=AUED codes, IEEE Trans. IT 39 (1993) 232–238.


Strong qualitative independence 

Repository of the Academy's Library

Strong qualitative independence∗†Gyula O.H. KatonaAlfre´d Re´nyi Institute of Mathematics, HAS,Budapest P.O.B. 127 H-1364 HUNGARYohkatona@renyi.huJanuary 26, 2012∗AMS Subject classification Primary 90B40, Secondary 05D05, 94A99, 94B99.Keywords: search, qualitatively independent sets, Hadamard matrix, code†The work was supported by the Hungarian National Foundation for ScientificResearch grant numbers T029255.AbstractThe subsetsA,B of the n-elementX are said to be s-strongly separating if thetwo sets divide X into 4 sets of size at least s. The maximum number h(n, s)of pairwise s-strongly separating subsets was asymptotically determined byFrankl [8] for fixed s and large n. A new proof is given. Also, estimates forh(n, cn) are found where c is a small constant.1 IntroductionLet X be a finite set of n elements. The following notion was introduced byMarczewski [17]. The subsets A,B ⊂ X are called qualitatively independentif they divide X into four non-empty parts that is A∩B,A∩B,A∩B,A∩Bare all non-empty. The significance of this notion in search theory lies in theconsequence that after knowing if an unknown x is in A or not we cannotdecide the same question for B, independenly on the answer for the firstquestion. The family F ⊂ 2X is called independent if their members arepairwise qualitatively independent.Re´nyi asked [23] the question what is the maximum size of an independentfamily in an n-element set. The answer was found by the present author ([12],see also [23]), and independently by Brace and Daykin [5], Bolloba´s [3] andKleitman and Spencer [16] (see also Scho¨nheim [24] ).Theorem 1.1 The maximum number of pairwise qualitatively independentset in an n-element sets is (n− 1⌈n2⌉ ).If the family F is independent then A,B ∈ F , A 6= B implies A 6⊂ Bthat is F is inclusion-free. The maximum size of an inclusion-free family isdetermined by a classical theorem of Sperner [25].Theorem 1.2 If F is an inclusion-free family in an n-element set then|F| ≤(n⌊n2⌋).The following sharpening of Sperner’s theorem is also well-known as theYBLM-inequality (earlier LYM) (see [27], [4], [13], [18]). It involves the sizesof the “levels” of the family. If F is a family, define fi(F) = |{F : F ∈F , |F | = i}|.Theorem 1.3 If F is an inclusion-free family in an n-element set thenn∑i=0fi(F)(ni) ≤ 1holds.1Kleitman and Spencer [16] introduced the notion of k-indpendence. Afamily F ⊂ 2X is called k-independent iff∩ki=1Aεii 6= ∅holds for any choice of distinct A1, . . . , Ak ∈ F and εi = 0, 1 where A0 =A,A1 = A that is when any k members divide X into 2k non-empty parts.Let f(n, k) denote the maximum size of a k-independent family. They provedthe following theorem.Theorem 1.4 (Kleitman and Spencer [16]).2c12−kk−1n ≤ f(n, k) ≤ 2c22−kn.See also [1].A set A can be considered as a partition (A,A). A generalization inthis direction is to consider partitions into r parts that is r-partitions. Twor-partitions are called qualitative independent if all the r2 intersections ofthe classes are non-empty. The maximum number of pairwise qualitativelyindependent r-partitions is denoted by g(n, r). As this is exponential in nand an exact formula for g(n, r) is hopeless, it is sufficient to consider theexponent:qr = lim supn→∞1nlog g(n, r).After some preliminary results ([19], [20], [21], [22], [15]) Gargano, Ko¨rnerand Vaccaro have determined the exact exponent.Theorem 1.5 (Gargano, Ko¨rner and Vaccaro [9]).qr =2r(2 ≤ r).For a recent interesting result concerning qualitative independence see[14] .A family S ⊆ 2X is s-strongly separating iff all four intersections A∩B,A∩B,A∩B,A∩B are of size at least s for any two distinct members A,B ∈ S.The maximum size of an s-strongly separating family is denoted by h(n, s).The determination of h(n, s) was asked in [12]. It has been asymptoticallyanswered by Frankl for fixed s.2Theorem 1.6 [8]d1(s)2nns−12≤ h(n, s) ≤ d2(s) 2nns−12.whered1(s) =√2pi12s− εandd2(s) =√2pi2s−2(s− 1)! + ε.The author returned to this question now because Benjamin Weiss [26]suggested the following related problem. Find the maximum number of se-quences of length n over the alphabet {0, 1, 2} such that if a = (a1, . . . , an), b =(b1, . . . , bn) are two such sequences then every pair (c, d) c, d ∈ {0, 1, 2} oc-curs either 0 times or at least twice among the pairs (ai, bi) (1 ≤ i ≤ n). Theanalogous problem for {0, 1} is just h(n, 2).However we only found a new proof for Frankl’s theorem and some esti-mates for the case when s = cn where c is a small constant.In the new proof of the upper estimate (in Theorem 1.6) it is sufficientto use the following weaker condition. A family F is called s-diffbounded iff|A−B| ≥ s holds for any two members of F . Let us mention that a family is1-diffbounded iff A 6⊂ B for any two distinct members, in other words, iff thefamily is inclusion-free. It is obvious that if S is s-strongly separating then itis s-diffbounded. The following theorem is a generalization of Theorem 1.3.Theorem 1.7 Let s be a positive integer. If F is an s-diffbounded family inan n-element set thenn∑i=s−1fi(F)(is−1)(ni−s+1) ≤ 1. (1.1)Christian Bey [2] called the author’s attention to the fact that Theorem1.7 is a special case of Theorem 4 of [28] (i = t in its notation).Corollary 1.8 Let s be a positive integer. If F is an s-diffbounded familyin an n-element set then|F| ≤ d3(s) 2nns−123whered3(s) =(√2pi2s−1(s− 1)! + ε).If S is s-strongly separating then the familyF = {S, S : S ∈ S}is s-diffbounded. Here |S| ≤ 12|F| holds and this yields the following im-provement of Corollary 1.8.Corollary 1.9 Let s be a positive integer. If S is an s-strongly separatingfamily in an n-element set then|S| ≤(√2pi2s−2(s− 1)! + ε)2nns−12.In Section 3 the case when s = cn is considered. We have a non-trivialupper estimate only when c is small.Theorem 1.10h(n, cn) ≤ 2n(− 12 log c−1.099)+o(n)where c ≤ 0.099.2 The case of fixed sProof of Theorem 1.7Define the r-shadow of a family F in the following way.σr(F) = {A−B : B ⊂ A ∈ F , |B| = r}.Let A,B ∈ F be distinct members and choose an s− 1-element subset Cof A. Here |A−B| ≥ s implies A−C 6⊆ B. If D is an s−1-element subset ofB then A−C 6⊆ B−D follows. This is obviously true if A = B but C 6= D.Therefore σs−1(F) is inclusion-free.Determine the sizes of the levels of σs−1(F). The sets on level j = i− s+1 (s− 1 ≤ i ≤ n) are obtained from sets on level i in F :fj(σs−1(F)) = fi(F)(is− 1).The YBLM-inequality for the family σs−1(F) is exactly (1.1).4Proof of Corollary 1.8Theorem 1.7 will be used. Find the minimum of the ratio(is−1)(ni−s+1)in the terms of (1.1). The above ratio is equal to(n+s−1s−1)(n+s−1i) .Its minimum is (n+s−1s−1)( n+s−1bn+s−12 c) .Therefore (1.1) implies|F| =n∑i=s−1fi(F) ≤( n+s−1bn+s−12 c)(n+s−1s−1) .The application of the Stirling formula leads to the desired result.It is easy to see thatd3(s) =√2pi2s−1(s− 1)! + εfor large enough n.Now Corollary 1.9 easily follows.In the proof of the lower estimate of Theorem 1.6 the following result willbe used from coding theory. A codeword is a 0,1 sequence of length n. Wesay that the codeword has weight w if the number of 1s is w. The Hammingdistance of two codewords is the number of different digits. Let A(n, 2δ, w)is the maximum number of codewords of weight w with pairwise Hammingdistance at least 2δ.Theorem 2.1 (Graham and Sloane [10] ) Let q ≥ n be a prime power. Then(nw)qδ−1≤ A(n, 2δ, w).5Proof of of Theorem 1.6The upper estimate is a consequence of Corollary 1.9.The construction of the lower estimate is based on Theorem 2.1. Use thetheorem for n − s, 2s, bn2c − s. The codewords define, in an obvious way,subsets of an n−s-element set Y with pairwise differences at least s. Denotethis family by F ′. Add a disjoint set U to Y , where |U | = s. The extendedunderlying set is X = Y ∪ U . DefineF = {F ∪ U : F ∈ F ′},that is U is contained in all the sets. Since the difference of any two membersof F ′ have size at least s, the same holds for any two members of F , that is,F satisfies the conditions of the theorem. By Theorem 2.1 we have(n−sbn2c−s)qs−1≤ |F|. (2.1)By the prime number theorem there is a prime q satisfying n ≤ q ≤ n+o(n).This fact, (2.1) and the Stirling formula proves the lower estimate withd1(s) =√2pi12s− εfor large enough n.3 The case when s = cnThe other end.In the previous section the case when s is much smaller than n wascosidered. Suppose now that n is divisible by 4 and s = n4. Let S be an n4-strongly separating family. Then A,B ∈ S, (A 6= B) divide X into four partsof n4elements each. Associate a vector with coordinates 1,−1 with a memberA of S writing 1 in the ith position iff the ith element of X is in A. Denotethe vectors obtained in this way from the members of S by v1, v2, . . . vm. Itis easy to see that the inner product vivj is 0 for 1 ≤ i < j ≤ m. Let v0have 1s in each coordinate. Then v0vi = 0 also holds (1 ≤ i ≤ m). Thatis, v0, v1, . . . , vm are pairwise orthogonal vetors in an n-dimensional space.Then m ≤ n− 1 follows. The following little statement is obtained.6Proposition 3.1 Let n divisible by 4. Thenh(n,n4)≤ n− 1with equality iff there is a Hadamard matrix.Consider now the case when n = 4s+ t with a small positive integer t. Then|vivj| ≤ t must hold for all 0 ≤ i < j ≤ m. Since |vi| = n (0 ≤ i ≤ m)therefore the normed version of vi is ui =vi|vi| =vi√n. Their lengths are 1, onthe other hand |uiuj| ≤ tn holds for 0 ≤ i < j ≤ m. We arrived to a geometricproblem: what is the maximum number m of points on the surface of then-dimensional unit ball Bn (it is called the n−1-dimensional spherical spaceSn−1) if the angle ϕ between any two vectors determined by these pointssatisfies | cosϕ| ≤ tn.A slightly different version of this problem is widely studied: what isthe maximum number M(d, ϕ) of points on the surface of the d = (n − 1)-dimensional spherical space Sd if the angle between any two vectors deter-mined by these points is at least ϕ. For good surveys see [7] and [6] .In our case the vectors u0, u1, . . . , um,−u0,−u1, . . . ,−um possess the prop-erty that their pairwise angle is at most arccos td+1. Denote the maximumnumber of points P1, . . . , Pm in the d-dimensional spherical space such thatthe angle between any two of these points and their opposites is at least ϕby N(d, ϕ). The inequality 2N(d, ϕ) ≤ M(d, ϕ) is obvious. It does notalways hold with equality as the case d = 2, ϕ = arccos 13shows, sinceM(2, arccos 13) = 9 while N(2, arccos 13) = 4. We have proved the follow-ing lemma.Lemma 3.2h(4s+t, s) ≤ N(4s+t−1, arccos t4s+ t)−1 ≤ 12M(4s+t−1, arccos t4s+ t)−1.In order to obtain good estimates on h(4s + t, s) for fixed t and large s oneshould study the corresponding values of M(d, ϕ) and N(d, ϕ). E.g. it seemsto be not difficult to prove the inequality h(4s+1, s) ≤ 4s+1, but h(5, 1) = 4by Theorem 1.1 showing that the equality does not always hold.The middle case.Here we want to find h(n, cn) where c is a constant and n is large. Ouronly result is Theorem 1.10.7Proof of Theorem 1.10. This upper estimate is a trivial combination ofLemma 3.2 and a theorem of Kabatjanskiˇı and Levensˇtein [11] claimingM(d, ϕ) ≤ (1− cosϕ)− d2 2−0.099d++o(d) (ϕ ≤ 62.9974 . . .◦).We have no reasonable lower estimate. Theorem 2.1 does not help, sinceits estimate tends to 0 when s = cn and n tends to ∞.4 Open problemsRestriction with the pairwise entropy.Let us repeat a problem posed in [12]. If p = (p1, . . . , pm) is a probabilitydistribution then its entropy isH(p) =m∑i=1−pi log pi.Define the entropy of a pair of sets A,B ⊂ X as the entropy of the probabilitydistribution ( |A ∩B||X| ,|A ∩B||X| ,|A ∩B||X| ,|A ∩B||X|).What is the maximum number of subsets A1, . . . , Am of an n-element set Xif the entropy of any pair Ai, Aj (i 6= j) is at least ρ? The case ρ = 4 issolved by Proposition 3.1.Combining the problems.The combinations of the problems of Theorems 1.4, 1.5 and 1.6 give riseto many new problems.References[1] Alon, Noga, Explicit construction of exponential sized families of k-independent sets, Discrete Mat. 58(1986) 191-193.[2] Bey, Christian, personal communication.8[3] Bolloba´s, B., Sperner systems consisting of pairs of complementarysubsets, J. Combin. Theory Ser A 15(1973) 363-366.[4] Bolloba´s, B., On generalized graphs, Acta Math. Hungar 16(1965)447-452.[5] Brace, A. and Daykin, D.E. Sperner-type theorems for finite sets, inD.J.A. Welsch and D.R. Woodall, editors, Combinatorics, (Proc. Conf.Combinatorial Math., Math. Inst., Oxford, 1972, pp. 18-37, Inst. Math.Appl., Southend-on-Sea (1972).[6] Conway, J.H. and Sloane, N.J.A. Sphere Packings, Lattices andGroups, 2nd edition. Springer Verlag, New York, 1993.[7] Fejes To´th, G., Spherical Space, in Handbook of Computational Ge-ometry pp. 32-34.[8] Frankl, P, An extremal problem of coding type, Ars Combin 1(1976)53–55.[9] Gargano, L., Ko¨rner, J. and Vaccaro, U., Sperner capacities,Graphs and Combin. 9(1993) 31-56.[10] Graham, R.L. and Sloane, N.J.A., Lower bounds for ConstantWeight Codes, IEEE Trans. Inform. Theory 26(1980) 37-43.[11] Kabatjanskiˇi, G.A., Levensˇtein, V.I., Bounds for packings on asphere and in space, Problemy Peredachy Inf. 14(1)(1978) 3-25 andProblems of Inf. Trans. 14(1)(1978) 1-17.[12] Katona, G.O.H., Two applications (for search theory and truth func-tions) of Spener type theorems, Periodica Math. Hungar. 3(1973) 19-26.[13] Lubell, D., A short proof of Sperner’s lemma, J. combin. Theory1(1966) 299.[14] Ko¨rner, J. and Monti, A., Delta -Systmes and Qualitative(In)Dependence, J. Combin. Theory 99(2002) 75–84.[15] Ko¨rner, J. and Simonyi, G., A Sperner-type theorem and qualita-tive independence, J. Comb. Theory Ser A, 59)(1992) 90–1039[16] Kleitman, D. and Spencer, J., Families of k-indpendent sets, Dis-crete Math 6(1973) 255-262.[17] Marczewski, E., Inde´pendance d’ensembles et prolongement demesures, Colloq. Math. 1(1948) 122-132.[18] Mesˇalkin, L.D., A generalization of Sperner’s theoremon the numberof subsets of a finite set, Teor. Verojatnost. i Primen. 8(1963) 219-220(in Russian with German summary).[19] Poljak, S., Pultr, A. and Ro¨dl, V., On the dimension of Knesergraphs, in Algebraic Methods in Graph Theory, Szeged, Hungary, Coll.Soc. Math. J. Bolyai, pp. 631-646.[20] Poljak, S., Pultr, A. and Ro¨dl, V., On qualitatively independentpartitions and related problems, Discrete Appl. Math. 6(1983) 193-205.[21] Poljak, S. and Ro¨dl, V., Orthogonal partitions and covering ofgraphs, Czech Math. J. 30(1980) 475-485.[22] Poljak, S., Tuza, Z., On the maximu number of qualitatively inde-pendent partititons, J. Combin. Theory Ser A 51(1989) 111-116.[23] Re´nyi, A., Foundations of Probability,Wiley, New York, 1971.[24] Scho¨nheim, J. On a problem of Purdy related to Sperner systems,Canad. Math. Bull. 17(1974) 135-136.[25] Sperner, E., Ein Satz u¨ber Untermengen einer endlichen Menge, Math.Z. 27(1928) 544-548.[26] Weiss, Benjamin, personal communication.[27] Yamamoto, K., Logarithmic order of free distributive lattices, J. Math.Soc. Japan 6(1954) 347-357.[28] Zhang, Zhen, LYM-TYpe Inequalities for tEC/AUED Codes, IEEETrans. IT 39(1993) 232-238.10

Strong Qualitative Independence

file:///data/remote/core/dit/data/elsevier/pdf/3a0/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDE2NjIxOHgwMzAwMTkwMg==.pdf

Strong Qualitative Independence

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Repository of the Academy's Library

Elsevier - Publisher Connector