Stein-Lovasz 定理的推廣及其應用
成果報告
黃大原
中文摘要：
Stein-Lovasz 定理在論及覆蓋相關的組合問題時，提供一種具有演算法
本質的存在性充分條件；進而為某些具有組合意義的參數提供其上界。在
本論文理，我們針對 Stein - Lovasz 定理討論其推廣，給出其演算法；進而
針對(d,s, out of r, z]-disjunct 矩陣、 (k,m,c,n;z) - 選子的實驗次數的上界作討
論，得出其上界。
中文關鍵字：Stein-Lovasz 定理、組合覆蓋問題、組合參數的上界、Lovasz
局部引理
英文摘要：
The Stein-Lovasz theorem provides an algorithmic way to dealing with the
existence of good coverings and then deriving some bounds related to some
combinatorial structures. An extension of the classical Stein-Lovasz theorem is
given, followed by some applications for finding upper bounds of the sizes of
(d,s, out of r, z]-disjunct matries, (k,m,c,n;z) -selectors respectively and some
others.
Key words: Stein-Lovasz theorem, Combinatorial covering problems, upper
An Extension of Stein-Lov´
asz Theorem and Some
of its Applications
Tayuan Huang
Guang-Siang Lee
October 31, 2008
Abstract
The Stein-Lov´asz theorem provides an algorithmic way to dealing with the existence of good coverings and then deriving some bounds related to some combinatorial structures. An extension of the classical Stein-Lov´asz theorem is given, followed by some applications for finding upper bounds of the sizes of (d, s out of r; z]-disjunct matries, (k, m, c, n; z)-selectors respectively and some others.
1
Introduction
Let X be a finite set and Γ be a family of subsets of X. We denote by H = (X, Γ) the hypergraph having X as the set of vertices and Γ as the set of hyperedges. A subset T ⊆ X such that TTE 6= ∅
for any hyperedge E is called a cover of the hypergarph H. The minimum size of a cover of an hypergraph H is denoted by τ (H). An upper bound for τ (H) was given by Lov´asz [L75]:
τ (H) ≤ |X|
minE∈Γ|E|(1 + ln 4), where 4 = maxx∈X|{E : E ∈ Γ with x ∈ E}|.
An equivalent statement in terms of the block-point incidence matrices of the corresponding hypergraphs was given by Stein [S74] independently (see Theorem 3.1). It was called the Stein-Lov´asz Theorem in [CLZ96] while dealing with the covering problems in coding theory. The Stein-Lov´asz theorem was first used in dealing with the upper bounds for the sizes of (k, m, n)-selectors [BGV05]. Inspired by this work, it was also used in dealing with the upper bound for the sizes of (d, r; z]-disjunct matrices [CFH08]. Some more applications can also be found in [DZR08, DZ08]. The notion of (k, m, n)-selectors was first introduced by de Bonis et al. in [BGV05], followed by a generalization to the notion of (k, m, c, n)-selectors [B08, pp.87]. A further generalization of (k, m, c, n)-selectors will be given in Section 2.
Definitions of several properties over binary matrices are considered in Section 2 including (d, s out of r; z]-disjunct matries, (k, m, c, n; selectors for group testing purpose, and uniform (m, t; z)-splitting systems, uniform (m, t1, t2; separating systems, (v, k, t; covering designs, (v, k, t, p; z)-lotto designs for others. Some formulas are given in Lemmas 2.1 ∼ 2.4 for later simplification purpose used in Section 4.
In order to dealing with the upper bounds for these binary matrices defined in Section 2, an extended Stein-Lov´asz Theorem is derived in Section 3. Some applications of the determination of some upper bounds of the sizes of various models are considered in Section 4. In the first half of Section 4, the extended Stein-Lov´asz theorem will be used in dealing the upper bound for the sizes of (d, s out of r; z]-disjunct matrices (Theorem 4.1) and (k, m, c, n; z)-selectors (Theorem 4.2). It in turn provides a few upper bounds for the sizes in various models with specific parameters as shown in Corollaries 4.1.1 ∼ 4.2.1. Those upper bounds for the sizes of uniform splitting systems, uniform separating systems, cover designs and lotto designs are given in Theorems 4.3 ∼ 4.6 respectively.
Most of these bounds are roughly the same as those derived by the basic probabilistic method including the Lov´asz Local Lemma. Thus, due to its constructive nature, the Stein-Lov´asz theorem can be regarded as a de-randomized algorithm for the probabilistic methods. The relationship between the extended Stein-Lov´asz theorem and the Lov´asz Local Lemma deserve further study [DZ08].
2
Preliminaries
2.1
A few models for various purposes
A few types of binary matrices will be introduced in this section, followed by corresponding associ-ated parameters. These families of binary matrices will be used as models for pooling designs and for many other identification purposes.
Definition 2.1
A binary matrix M of t × n order is called (d, s out of r; z]-disjunct, 1 ≤ s ≤ r, if for any d columns and any other r columns of M , there exist z row indices in which none of the d columns appear and at least s of the r columns do. The integer t is called the size of the (d, s out of r; z]-disjunct matrix. The minimum number of rows over all (d, s out of r; z]-disjunct matrices with n columns is denoted by t(n, d, r, s; z].
Definition 2.2
For integers k, m, s and n with 1 ≤ s ≤ k ≤ n and 1 ≤ m ≤ ¡ks¢, a t × n binary matrix M is called a (k, m, c, n; z)-selector if any t × k submatrix of M contains at least z disjoint submatrices with each row weight exactly s, with at least m distinct rows each. The integer t is called the size of the (k, m, c, n; z)-selector. The minimum size over all (k, m, c, n; z)-selectors is denoted by
ts(k, m, c, n; z).
It is interesting to remark that the notion of (k, m, n)-selectors, first introduced by de Bonis et. al. in [BGV05] and it was then generalized to the notion of (k, m, c, n)-selectors [B08, pp.87], which are equivalent to (k, m, 1, n; 1)-selectors and (k, m, w, n; 1)-selectors rsepectively. The upper bounds for the size of (k, m, n)-selectors and (k, m, c, n)-selectors were studied in [BGV05] and in [B08, pp. 87 ∼ 91] respectively by the Stein-Lov´asz theorem. The bound for the size of (k, m, c, n; z)-selectors will be derived by the extended Stein-Lov´asz theorem (Theorem 3.2) in Theorem 4.2. The following are tables of subclasses of (d, s out of r; z]-disjunct matrices and of (k, m, c, n; z)-selectors respectively.
parameters types bounds references
s = r = 1, z = 1 d-disjunct t(d, n) [Y02] s = r = 1 de-disjunct s = r, z = 1 (d, r]-disjunct t(n, d, r] [Y07] s = r (d, r; z]-disjunct t(n, d, r; z] [CFH08] s = 1, z = 1 (d, r)-disjunct t(n, d, r) [Y07] s = 1 (d, r; z)-disjunct t(n, d, r; z)
z = 1 (d, s out of r]-disjunct t(n, d, r, s] [Y07]
(d, s out of r; z]-disjunct t(n, d, r, s; z]
parameters types bounds references
c = 1, z = 1 (k, m, n)-selectors ts(k, m, n) [BGV05,Y07]
c = 1 (k, m, n; z)-selectors ts(k, m, n; z)
z = 1 (k, m, c, n)-selectors ts(k, m, c, n) [B08]
(k, m, c, n; z)-selectors ts(k, m, c, n; z)
As the relationship between various models (disjunct matrices, selectors) and nonadaptive group testing:
1. A (d, r]-disjunct matrix can be used to identify the up-to-d positives on the complex model (ref to Chen, Du and Hwang [CDH07]).
2. A (h, d)-disjunc is a necessary condition for identifying the positive set on the (d, h)-inhibitor model [BV98].
3. There exists a two-state group testing algorithm for finding up-to-d positives out of n items and that uses a number of tests equal to t + k − 1, where t is the size of a (k, d + 1, n)-selector [BGV05].
Definition 2.3
Let m and t be even integers with 2 ≤ t ≤ m, and let z be a positive integer. An uniform (m, t; z)-splitting system is a pair (X, Γ) where |X| = m, Γ is a set of m
2–subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exist at least z blocks B ∈ Γ such that |TTB| = t
2, i.e., B splits T . The system (X, Γ) is also called a (t; z)-splitting system. The minimum number of blocks over all (t; z)-splitting systems is denoted by SP (m, t; z).
Definition 2.4
Let m be an even integer and let t1, t2 and z be positive integers with t1+ t2 ≤ m. An uniform (m, t1, t2; z)-separating system is a pair (X, Γ) where |X| = m, Γ is a set of m2–subsets of X, called blocks such that for every T1, T2 ⊆ X , where |Ti| = ti for i = 1, 2 and |T1
T
T2| = ∅, there exist at least z blocks B ∈ Γ for which either T1 ⊆ B, T2
T
B = ∅ or T2 ⊆ B, T1 T
B = ∅, i.e., P , Q
are separated by B. The system (X, Γ) is also called a (t1, t2; z)-separating system. The minimum number of blocks over all (t1, t2; z)-separating systems is denoted by SE(m, t1, t2; z).
Definition 2.5
Let v, k, t and z be positive integers with t ≤ k ≤ v. A (v, k, t; z)-covering design is a pair (X, Γ) where |X| = v, Γ is a family of k-subsets of X, called blocks such that for every T ⊆ X with
|T | = t, there exist at least z blocks B ∈ Γ containing T . The minimum number of blocks over all
(v, k, t; z)-covering designs is denoted by C(v, k, t; z). Definition 2.6
Let v, k, t, p and z be positive integers with p ≤ t, k ≤ v. A (v, k, t, p; z)-lotto design is a pair (X, Γ) where |X| = v, Γ is a set of k-subsets of X, called blocks such that for every T ⊆ X with |T | = t, there exist at least z blocks B ∈ Γ such that TTB ≥ p. The minimum number of blocks over all
(v, k, t, p; z)-lotto designs is denoted by L(v, k, t, p; z).
Note that when p = t, a (v, k, t, t; z)-lotto design will be reduced to a (v, k, t; z)-covering design. The related bounds are summarized in the following table.
types bounds references uniform (m, t; z)-splitting systems SP (m, t; z) z = 1[DZR08]
uniform (m, t1, t2; z)-separating systems SE(m, t1, t2; z) z = 1[DZR08] (v, k, t; z)-covering designs C(v, k, t; z) z = 1[DZR08]
(v, k, t, p; z)-lotto designs L(v, k, t, p; z) z = 1[DZR08]
2.2
Some basic counting results
Stein-Lov´asz Theorem and its extension will be used to estimate the upper bounds of the sizes for pooling designs of various models. In order to give upper bounds for the above mentioned parameters, the following results involving binomial coefficients will be involved. Lemma 2.1 will be used in showing appropriate values of w for pooling designs of various models. We need information regarding the maximum of the function
f (w) = µ n − w d ¶µ n − w − d r − s ¶µ w s ¶
with various r and s when dealing with possible upper bounds for t of various models. Lemmas 2.2
∼ 2.4 will be used in the simplifications of the bounds M
v, and ln a respectively in the expression M
Since ex = X n≥0
xn
n!, we have ex ≥ xb
b! for each x, thus eb ≥ bb b!, i.e., 1 ≤ b!eb bb ; and hence µ a b ¶ = a! (a − b)!b! = a(a − 1) · · · (a − b + 1) b! ≤ ab b! ≤ ab b! · b!eb bb = ( ea b ) b, i.e., µ a b ¶ ≤ a b b! ≤ ( ea b ) b. Lemma 2.1
For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, the function f (w) = µ n − w d ¶µ n − w − d r − s ¶µ w s ¶
gets its maximum at w =ns − (k − s)
k .
Proof. First we note that f (w) = µ n − w d ¶µ n − w − d r − s ¶µ w s ¶ = (n − w)! (n − w − d)!d!· (n − w − d)! (n − w − d − r + s)!(r − s)!· µ w s ¶ ·(d + r − s)! (d + r − s)! = µ n − w d + r − s ¶µ w s ¶µ d + r − s d ¶ = µ n − w k − s ¶µ w s ¶µ k − s d ¶ . Since f (w + 1) = µ n − (w + 1) k − s ¶µ w + 1 s ¶µ k − s d ¶ = ((n − w) − (k − s) n − w µ n − w k − s ¶ ) · ( w + 1 w + 1 − s µ w s ¶ ) · µ k − s d ¶ = ((n − w) − (k − s) n − w · w + 1 w + 1 − s)f (w), and (n − w) − (k − s) n − w · w + 1 w + 1 − s = (w + 1)(n − w) − (w + 1)(k − s) (w + 1)(n − w) − s(n − w) = 1 if and only if s(n − w) = (w + 1)(k − s), i.e., w = ns − (k − s)
k ; hence (n − w) − (k − s) n − w · w + 1 w + 1 − s ≥ 1 for w ≤ ns − (k − s) k and (n − w) − (k − s) n − w · w + 1 w + 1 − s ≤ 1 for w ≥ ns − (k − s) k .
As a consequence, we then have
f (w) is increasing for w ≤ ns − (k − s) k , and f (w) is decreasing for w ≥ ns − (k − s) k as required. By taking s = r = c in f (w) = µ n − w d ¶µ n − w − d r − s ¶µ w s ¶
, we get the quadratic function
g(w) = µ w c ¶µ n − w k − c ¶
Corollary 2.1.1 The function g(w) = µ w c ¶µ n − w k − c ¶
gets its maximum at w =nc − (k − c)
k .
Lemma 2.2
For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r,
n(n − 1) · · · (n − s + 1)(n − s)(n − s − 1) · · · (n − r − d + 1) n(n −k
s) · · · (n − k + ks) · n · (n − k−sk ) · · · (n − k + k−sk )
≤ 1. Proof. Without loss of generality, let s ≤ k − s and thus 1 ≤ k
k−s ≤ 2 ≤ ks. Moreover, we note that the left hand side is
n(n − 1) · · · (n − s + 1)(n − s)(n − s − 1) · · · (n − r − d + 1) n(n −ks) · · · (n − k + ks) · n · (n −k−sk ) · · · (n − k +k−sk ) = Y 0≤i≤r+d−1 (n − i) Y 0≤i≤s−1 (n − i ·k s) Y 0≤j≤k−s−1 (n − j · k k − s) .
To prove this inequality, we will rearrange the terms in the denominator so that n − t
n − f (t) ≤ 1 for
each t with 0 ≤ t ≤ r + d − 1, i.e., we will give a bijection
f : {0, 1, ..., r + d − 1} → {i ·k
s | 0 ≤ i ≤ s − 1}
[
{j · k
k − s| 0 ≤ j ≤ k − s − 1}
with the property that f (t) ≤ t for each t. Note that the element 0 will be counted twice as 0 ·k s and 0 · k
k−s respectively in the range of the function f . Note also that if i ·ks = i + i ·k−sk > t, then s
k−s< t−ii and hence j ·k−sk = j(1 +k−ss ) < j(1 +t−ii ) = j(t−ii ) = j(ji), where i + j = t. A such function f is defined recursively as follows. For t = 0, 1, 2, let f (0) = 0k
s, f (1) = 0k−sk ,
f (2) = k
k−s. For 3 ≤ t ≤ r + d − 1, let i (resp. j) be the smallest positive integers such that i ·ks (resp. j · k
k−s) /∈ {f (0), f (1), ..., f (t − 1)} if they exist, it follows that t = i + j. 1. Let f (t) = i · k
s if i ·ks ≤ t. 2. Otherwise, we define f (t) = j · k
k−s.
3. Finally, suppose t is large and there is no such i, note that
n − s n ≤ n − s − 1 n − k k−s ≤ · · · ≤ n − r − d + 1 n − k + k k−s and n − r − d + 1 n − k + k k−s < 1, we have n − t n − (t − s) k k−s
≤ 1 for all s ≤ t ≤ r + d − 1, we define f (t) = (t − s) k k−s.
Cearly, the above defined function is 1-1, onto, and f (t) ≤ t for all 0 ≤ t ≤ r + d − 1 as required.
Lemma 2.3
For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, let n0 ≥ n be the smallest positive integer such that w =n0s
k is an integer, then µ n0 d ¶µ n0− d r ¶ µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ ≤ (k s) s( k k − s) k−s µ r s ¶ .
Proof. µ n0 d ¶µ n0− d r ¶ µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ = (n0)! (n0− d)!d!· (n0− d)! (n0− d − r)!r! (n0− w)! (n0− w − d)!d!· (n0− w − d)! (n0− w − d − r + s)!(r − s)!· w! (w − s)!s! = 1 r!n0(n0− 1) · · · (n0− s + 1)(n0− s)(n0− s − 1) · · · (n − d − r + 1) 1 (r−s)!s!w(w − 1) · · · (w − s + 1)(n0− w)(n0− w − 1) · · · (n0− w − d − r + s + 1) =¡1r s ¢ · n0s n0(n0− 1) · · · (n0− s + 1)(n0− s)(n0− s − 1) · · · (n − d − r + 1) k (n 0s k − 1) · · · (n 0s k − s + 1)(n0−n 0s k )(n0−n 0s k − 1) · · · (n0−n 0s k − d − r + s + 1) =¡1r s ¢ · 1 (s k)s(k−sk )k−s ·n 0(n0− 1) · · · (n0− s + 1)(n0− s)(n0− s − 1) · · · (n − d − r + 1) n0(n0−k s) · · · (n0− k +ks) · n0· (n0−k−sk ) · · · (n0− k + k−sk ) = (k s) s( k k − s) k−s µ r s ¶ (by Lemma 2.2). Lemma 2.4
For any positive integers n, d, r, s with k = d + r ≤ n and 1 ≤ s ≤ r, let n0 ≥ n be the smallest positive integer such that w =n
0s k is an integer, then ln( µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ ) < k[1 + ln(n k+ 1)] + ln µ k − s d ¶ . Proof. First we note that n0< n + k for such n0. Since
µ a b ¶ ≤ (ea b ) b, we have µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ = (n 0− w)! (n0− w − d)!d!· (n0− w − d)! (n0− w − d − r + s)!(r − s)!· µ w s ¶ ·(d + r − s)! (d + r − s)! = µ n0− w d + r − s ¶µ w s ¶µ d + r − s d ¶ = µ n0− w k − s ¶µ w s ¶µ k − s d ¶ ≤(e(n 0−n0s k ) k − s ) k−s· (e(n 0s k ) s ) s· µ k − s d ¶ = (e ·n 0 k) k−s· (en0 k) s· µ k − s d ¶
= ek(n0 k) k· µ k − s d ¶ < ek(n + k k ) k· µ k − s d ¶ = ek(n k+ 1) k· µ k − s d ¶ . Therefore, ln( µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ ) < ln(ek(n k+ 1) k· µ k − s d ¶ ) = k[1 + ln(n k+ 1)] + ln µ k − s d ¶ .
The substitutions of w for various subclasses are summarized in the following table for later references: types parameters d-disjunct s = r = 1, z = 1 w = n − d k w = n0 k de-disjunct s = r = 1 w = n − d k w = n0 k (d, r]-disjunct s = r, z = 1 w = nr − d k w = n0r k (d, r; z]-disjunct s = r w = nr − d k w = n0r k (d, r)-disjunct s = 1, z = 1 w = n − (k − 1) k w = n0 k (d, r; z)-disjunct s = 1 w = n − (k − 1) k w = n0 k (d, s out of r]-disjunct z = 1 w = ns − (k − s) k w = n0s k (d, s out of r; z]-disjunct w = ns − (k − s) k w = n0s k types parameters (k, m, n)-selectors c = 1, z = 1 w = n − (k − 1) k w = n0 k (k, m, n; z)-selectors c = 1 w = n − (k − 1) k w = n0 k (k, m, c, n)-selectors z = 1 w = nc − (k − c) k w = n0c k (k, m, c, n; z)-selectors w = nc − (k − c) k w = n0c k
3
The Stein-Lov´
asz Theorem and its extension
We now introduce the Stein-Lov´asz Theorem as follows. The proof is included for completeness [DZR08]. The Stein-Lov´asz theorem can be further extended from rows with weight at least 1 to the case of rows with weight at least z ≥ 1. The bound can be further improved when M is a matrix with constant row weight and column weight as well, i.e., in the language of hypergraphs, uniform and regular.
Theorem 3.1 [DZR08] Let A be a (0,1) matrix with N rows and M columns. Assume that each row contains at least v ones, and each column at most a ones. Then there exists an N × K submatrix C with K ≤ (N a) + ( M v ) ln a ≤ ( M v )(1 + ln a),
such that C does not contain an all-zero row.
Proof. A constructive approach for producing C is presented. Let Aa = A. We begin by picking the maximal number Ka of columns from Aa, whose supports are pairwise disjoint and each column having a ones (perhaps, Ka = 0). Discarding these columns and all rows incident to one of them, we are left with a ka× (M − Ka) matrix Aa−1, where ka = N − aKa. Clearly, the columns of Aa−1 have at most a − 1 ones (indeed, otherwise such a column could be added to the previously discarded set, contradicting its maximality). Now we remove from Aa−1 a maximal number Ka−1 of columns having a − 1 ones and whose supports are pairwise disjoint, thus getting a ka−1× (M − Ka− Ka−1) matrix Aa−2, where ka−1= N −aKa−(a−1)Ka−1. The process will terminate after at most a steps. The union of the columns of the discarded sets form the desired submatrix C with K =
a X i=1
Ki. The first step of the algorithm gives ka = N − aKa, which we rewrite, setting ka+1 = N , as
Ka = ka+1a−ka. Analogously, Ki = ki+1i−ki, i = 1, ..., a. Now we derive an upper bound for ki by counting the number of ones in Ai−1 in two ways: every row of Ai−1 contains at least v ones, and every column at most i − 1 ones, thus
vki≤ (i − 1)(M − Ka− · · · − Ki+1) ≤ (i − 1)M. Furthermore, K = a X i=1 Ki= a X i=1 ki+1− ki i = ka+1 a + ka a(a − 1)+ ka−1 (a − 1)(a − 2)+ · · · + k2 1 × 2 − k1
≤ (N/a) + (M/v)(1/a + 1/(a − 1) + · · · + 1/2),
thus giving the result.
The greedy procedure as shown in the proof constructs the desired submatrix one column at a time ([DZ08, pp.4]), and hence the algorithm below follows.
Algorithm: STEIN-LOV ´ASZ(A)
input: A is an N × M matrix, each column has at most a ones, each row has at least v ones
C ← ∅
while A has at least one row
do find a column c in A having maximum weight delete all rows of A that contain a “1” in column c delete column c from A
output: Returns a submatrix of A with no all-zero row
At each state, a new column is added to the submatrix that maximizes the number of “new” rows that are yet uncovered. When all rows are covered, the algorithm stops. It seems quite interesting that we can use the Stein-Lov´asz Theorem to derive bounds for some combinatorial array [DZ08,
pp.4]. If the entries of 1 occur in a binary matrix of order N × M as uniformly as possible, for example, v-uniform and a-regular, an upper bound for K is found so that the submatrices of order
N × K will share similar property too.
Theorem 3.2 (Extension of The Stein-Lov´asz Theorem)
Let A be a (0,1) matrix of order N × M , and let a, v, z be positive integers. Assume that each row contains exactly v ones, and each column exactly a ones. Then there exists a submatrix C of order
N × K with
K ≤ z(z + 1)
2 (
M
v )(1 + ln a),
such that each row of C has weight at least z. More specifically, if the matrix is v-uniform and
a-regular, the upper bound can be reduced to K ≤ z(M
v )(1 + ln a).
The strategy for the proof of Theorem 3.2 is as follows:
1. Use the Stein-Lov´asz Theorem to obtain a submatrix C1 with each row weight at least 1. 2. Choose some columns in the matrix A\C1 to combine with the submatrix C1 to form a
sub-matrix C2 with each row weight at least 2.
3. Choose some columns in the matrix A\C2 to combine with the submatrix C2 to form a sub-matrix C3 with each row weight at least 3.
4. Step by step, and finally we obtain the desired submatrix C = Cz with each row weight at least z.
Note that this upper bound makes sense only ifz(z + 1) 2 ( M v )(1+ln a) < M , i.e., z(z + 1) 2 < v 1 + ln a in general, or if z(M v)(1 + ln a) < M , i.e., z < v
1 + ln a for the case of uniform and regular.
Proof. A constructive approach for producing C is presented. Let A1 = A. By the Stein-Lov´asz Theorem, there exists an N × M1submatrix C1(= B
0
1= B1) of A1with M1≤ Mv(1 + ln a) such that each row of C1 has weight at least 1.
The algorithm used in the proof of the Stein-Lov´asz Theorem shows that some rows of C1have weight exactly 1. Let R1be the set of indices of those rows and |R1| = r1. Let A2be the submatrix of order r1× (M − M1) obtained from A1 by deleting the submatrix C1and the i-th row, i /∈ R1as well. Then each row of A2 contains at least v − 1 ones, and each column at most a ones. Again, by the Stein-Lovsz Theorem, there exists an r1× M2 submatrix B2 with M2 ≤ M −Mv−11(1 + ln a) such that each row of B20 has weight at least 1. Let B2 be the matrix of order N × M2 obtained from
B20 by adding the i-th row, i /∈ R1. Let C2 be the matrix of order N × (M1+ M2) obtained by the union of B1and B2. Then C2 is a submatrix of A, and each row of C2has weight at least 2.
Similarly, there exist some rows of C2that have weight exactly 2. Let R2be the set of indices of those rows and |R2| = r2. Continue in this way, we have:
For 2 ≤ i ≤ z,
1. Ai is a matrix of order ri−1× (M − i−1 X j=1
Mj), and each row contains at least v − (i − 1) ones, and each column at most a ones.
2. Bi0 is an ri−1× Misubmatrix of Aiwith Mi ≤
M −
i−1 X j=1
Mj
v − (i − 1) (1 + ln a), and each row has weight
For 1 ≤ i ≤ z, 3. Bi is a matrix of order N × Mi. 4. Ci is an N × i X j=1
Mj submatrix of A, and each row has weight at least i.
Hence, C = Cz is the submatrix required, that is,
K = z X j=1 Mj= M1+ M2+ · · · + Mz ≤M v (1 + ln a) + M − M1 v − 1 (1 + ln a) + · · · + M − z−1 X j=1 Mj v − (z − 1) (1 + ln a) ≤M v (1 + ln a) + M v − 1(1 + ln a) + · · · + M v − (z − 1)(1 + ln a) = M (1 + ln a)(1 v + 1 v − 1+ · · · + 1 v − (z − 1)) ≤ M (1 + ln a)(1 v + 2 v + · · · + z v) =z(z + 1) 2 ( M v )(1 + ln a),
thus gives the result.
More specifically, for the case of uniform and regular, using similar argument as above with a minor modification. First we note that N v = M a by counting the number of entries 1 in A in two ways. For 2 ≤ i ≤ z , Ai is a matrix of order ri−1× (M −
i−1 X j=1
Mj), and each row contains exactly
v − (i − 1) ones, and each column at most a ones. Moreover, a lower bound for
i−1 X j=1
Mj is derived by counting the number of 1s in the submatrix Ci−1in two ways; each row of Ci−1contains at least
i−1 ones, and each column exactly a ones, thus N (i−1) ≤ (
i−1 X j=1
Mj)a, and henceM
v (i−1) ≤ i−1 X j=1 Mj for 2 ≤ i ≤ z. Furthermore, K = z X j=1 Mj = M1+ M2+ · · · + Mz ≤M v (1 + ln a) + M − M1 v − 1 (1 + ln a) + · · · + M − z−1 X j=1 Mj v − (z − 1) (1 + ln a) ≤M v (1 + ln a) + M −M v v − 1 (1 + ln a) + · · · + M − (z − 1) ·M v v − (z − 1) (1 + ln a) =M v (1 + ln a) + M v (1 + ln a) + · · · + M v (1 + ln a) = z(M v )(1 + ln a),
For the case of uniform and regular, Theorem 3.2 is restated in the language of hypergraphs in the following corollary. A z-cover is a subset C of X such that |CTE| ≥ z for any hyperedges E.
Let τz(H) be the minimum size over all z-covers of the hypergraph H.
Corollary 3.3 Let H = (X, Γ) be a v-uniform and a-regular hypergraph with vertex set X and edge set Γ, then τz(H) ≤ z
|X|
v (1 + ln a).
We conjecture that τz(H) ≤ zτ1(H) holds for hypergraphs which are uniform and regular. How-ever, it need not be true in general as shown in the following example. For the hypergraph with where X = {1, 2, 3, ..., 8} and Γ = {{1, 2, 3}, {4, 5, 6}, {1, 7, 8}}. It is easy to see that {1, 4} is a 1-cover with minimum size, hence τ1(H) = 2. Similarly, {1, 2, 4, 5, 7} is a 2-cover with minimum size, hence τ2(H) = 5. This shows that τ2(H) = 5 > 2 · 2 = 2τ1(H).
4
Some Applications of the extended Stein-Lov´
asz Theorem
The Stein-Lov´asz theorem was first used in dealing with the upper bound for the sizes in the model of (k, m, n)-selecters [BGV05]. Inspired by this work, it was also used in dealing with the upper bound for the sizes of (d, r; z]-disjunct matrices [CFH08]. In the first half of Section 4, the extended Stein-Lov´asz theorem will be used in dealing the upper bound for the sizes of (d, s out of r; z]-disjunct matrices (Theorem 4.1) and (k, m, c, n; z)-selectors (Theorem 4.2). It in turn provides a few upper bounds for the sizes in various models with specific parameters as shown in Corollaries 4.1.1 ∼ 4.2.1. Those upper bounds for the sizes of uniform splitting systems, uniform separating systems, cover designs and lotto designs are given in Theorems 4.3 ∼ 4.6 respectively.
4.1
bounds for various models of pooling designs
Upper bounds for the sizes of d-disjunct matrices, (d, r]-disjunct matrices, (d, r)-disjunct matrices and (d, s out of r]-disjunct matrices were given in [Y02, Y07] Lov´asz local lemma. Recall that
t(n, d, r, s; z] is the minimum of the sizes of (d, s out of r; z]-disjunct matrices.
Theorem 4.1
For any positive integers n, d, r, s and z, with 1 ≤ s ≤ r, if k = d + r ≤ n, then
t(n, d, r, s; z] < z( k s)s(¡k−sk )k−s r s ¢ {1 + k[1 + ln(n k + 1)] + ln µ k − s d ¶ }.
Proof. For s ≤ w ≤ n − d, let A be the binary matrix of order [
µ n d ¶µ n − d r ¶ ] × µ n w ¶ with rows indexed by {(D, R) | D ∈ µ [n] d ¶ , R ∈ µ [n] r ¶
with DTR empty}, and the columns indexed by V = {v | v ∈ {0, 1}n, wt(v) = w}. The entry of A at the row indexed by the pair (D, R) and the
column indexed by the vector v ∈ V is 1 if the entries of v over D are all zero and at least s entries of v over R are one; and 0 otherwise.
Observe that each row of A has weight
min(r,w)X j=s µ r j ¶µ n − (d + r) w − j ¶
, and each column of A has
weight µ n − w d ¶min(r,w)X j=s µ n − w − d r − j ¶µ w j ¶
exists a submatrix M of A of order [ µ n d ¶µ n − d r ¶
] × t with each row weight at least z, where
t ≤ z µ n w ¶ min(r,w)X j=s µ r j ¶µ n − (d + r) w − j ¶ {1 + ln[ µ n − w d ¶min(r,w)X j=s µ n − w − d r − j ¶µ w j ¶ ]} = z µ n d ¶µ n − d r ¶ µ n − w d ¶min(r,w)X j=s µ n − w − d r − j ¶µ w j ¶ {1 + ln[ µ n − w d ¶min(r,w)X j=s µ n − w − d r − j ¶µ w j ¶ ]}.
Note that the equality is obtained by counting the number of 1s in A in two ways. It is straightforward to show that the columns of M form a (d, s out of r; z]-disjunct matrix of order t × n. We then have
t(n, d, r, s; z] ≤ z µ n d ¶µ n − d r ¶ µ n − w d ¶min(r,w)X j=s µ n − w − d r − j ¶µ w j ¶ {1 + ln[ µ n − w d ¶min(r,w)X j=s µ n − w − d r − j ¶µ w j ¶ ]} ≤ z µ n d ¶µ n − d r ¶ µ n − w d ¶µ n − w − d r − s ¶µ w s ¶ {1 + ln[ µ n − w d ¶µ n − w − d r − s ¶µ w s ¶ ]}(only take j = s).
For given positive integers n, d, r and s, let n0 ≥ n be the smallest positive integer such that n0s
k is an integer. We have µ n0 d ¶µ n0− d r ¶ µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ ≤ (k s) s( k k − s) k−s µ r s ¶ by Lemma 2.3, and ln( µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ ) < k[1 + ln(n k+ 1)] + ln µ k − s d ¶ by Lemma 2.4. Therefore, t(n, d, r, s; z] ≤ t(n0, d, r, s; z] ≤ z µ n0 d ¶µ n0− d r ¶ µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ {1 + ln[ µ n0− w d ¶µ n0− w − d r − s ¶µ w s ¶ ]} < z( k s)s(¡k−sk )k−s r s ¢ {1 + k[1 + ln(n k + 1)] + ln µ k − s d ¶ } as required.
As a consequence, upper bounds for the sizes of various situations are summarized in the follow-ing corollary.
Corollary 4.1.1 1. If k = d + 1 ≤ n, then t(d, n) = t(n, d, 1, 1; 1] < k(k d) d{1 + k[1 + ln(n k + 1)]}. 2. If k = d + r ≤ n, then t(n, d, r] = t(n, d, r, r; 1] < (k r) r(k d) d{1 + k[1 + ln(n k+ 1)]}. t(n, d, r; z] = t(n, d, r, r; z] < z(k r) r(k d) d{1 + k[1 + ln(n k + 1)]}.[CFH08] t(n, d, r) = t(n, d, r, 1; 1] <k r(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln µ k − 1 d ¶ }. t(n, d, r; z) = t(n, d, r, 1; z] < zk r(1 + 1 k − 1) k−1{1 + k[1 + ln(n k+ 1)] + ln µ k − 1 d ¶ }. t(n, d, r, s] = t(n, d, r, s; 1] <( k s)s(¡k−sk )k−s r s ¢ {1 + k[1 + ln(n k + 1)] + ln µ k − s d ¶ }.
The notion of (k, m, n)-selectors was first introduced by de Bonis et. al. in [BGV05] , and it was then generalized to the notion of (k, m, w, n)-selectors [B08, pp.87]. It is interesting to remark that the notions of (k, m, n)-selecters and (k, m, w, n)-selectors are equivalent to (k, m, 1, n; 1)-selectors and (k, m, w, n; 1)-selectors respectively.
An upper bound for the size of (k, m, n)-selectors was also given in [Y07, 4.4.1] by Lov´asz local lemma.
Using similar arguments in [BGV05] and [B08] with a minor modification, the upper bound of the size of (k, m, c, n; z)-selectors is given below.
Theorem 4.2 For a = µ k c ¶ , ts(k, m, c, n; z) < (a − m + 1)(z − 1) + 1 a − m + 1 ( k c) c(1 + 1 k − c) k−c{1 + k[1 + ln(n k + 1)] + ln µ a − 1 a − m ¶ }. Proof. For c ≤ w ≤ n − k + c, let X = {x ∈ {0, 1}n| wt(x) = w} and U = {u ∈ {0, 1}k | wt(u) = c}.
Moreover, for any A ⊆ U of size r, r = 1, ...,¡kc¢, and any set S ∈ ¡[n]k¢, define EA,S = {x ∈ X :
x|S ∈ A}.
Let a =¡kc¢and M be a (0,1) matrix of order [¡a−m+1a ¢¡nk¢] סwn¢with rows indexed by the sets in Γ = {EA,S ⊆ X | A ⊆ U with |U | = a, |A| = a − m + 1, S ∈
¡[n] k
¢
}, and the columns indexed by
the vectors in X = {x ∈ {0, 1}n| wt(x) = w}. The entry of M at the row indexed by the set EA,S and the column indexed by the vector x ∈ X is 1 if x ∈ EA,S; and 0 otherwise.
Observe that each row of M has weight ¡a−m+11 ¢¡n−kw−c¢, and each column of M has weight ¡w c ¢¡n−k w−c ¢¡ a−1 (a−m+1)−1 ¢
. By the extension of the Stein-Lov´asz Theorem, there exists a submatrix M of M of order [¡a−m+1a ¢¡nk¢] × t with each row weight at least (a − m + 1)(z − 1) + 1, where
t ≤ [(a − m + 1)(z − 1) + 1] ¡n w ¢ (a − m + 1)¡n−kw−c¢ {1 + ln[ µ w c ¶µ n − k w − c ¶µ a − 1 a − m ¶ ]} = [(a − m + 1)(z − 1) + 1] ¡ a a−m+1 ¢¡n k ¢ ¡w c ¢¡n−w k−c ¢¡a−1 a−m ¢ {1 + ln[ µ w c ¶µ n − k w − c ¶µ a − 1 a − m ¶ ]}.
It suffices to show that the matrix M∗of order t×n formed by the columns of M is a (k, m, c, n; z)-selector, that is, any submatrix of k arbitrary columns of M∗contains at least z disjoint submatrices with each row weight exactly c, with at least m distinct rows each.
Let x1, x2, ..., xt be the t rows of M∗ and let T = {x1, x2, ..., xt}. Suppose contradictorily that there exists a set S ∈¡[n]k¢ such that the submatrix M∗|
S of M∗ contains at most z − 1 disjoint submatrices with each row weight exactly c, with at least m distinct rows each. Moreover, M∗|
S contains another disjoint submatrix with at most m − 1 distinct rows with weight exactly c. Let
uj1, uj2, ..., ujqbe such rows, with q ≤ m−1; let A be any subset of U \{uj1, uj2, ..., ujq} of cardinality
|A| = a − m + 1, then we have |TTEA,S| ≤ (a − m + 1)(z − 1) + 1, contradicting the fact that M is a matrix of order [¡ a
a−m+1 ¢¡n
k ¢
] × t with each row weight at least (a − m + 1)(z − 1) + 1. Hence we have ts(k, m, c, n; z) ≤ [(a − m + 1)(z − 1) + 1]¡a−m+1a ¢¡nk¢ ¡w c ¢¡n−w k−c ¢¡a−1 a−m ¢ {1 + ln[ µ w c ¶µ n − k w − c ¶µ a − 1 a − m ¶ ]}.
This bound can be simplified further as shown below. Let n0≥ n be the smallest positive integer such that w = n0c k is an integer. We have ¡ a a−m+1 ¢¡n0 k ¢ ¡w c ¢¡n0−w k−c ¢¡a−1 a−m ¢ = a! (m−1)!(a−m+1)! (a−1)! (m−1)!(a−m)! · ¡n0 k ¢ ¡w c ¢¡n0−w k−c ¢ = 1 a − m + 1· a¡n0 k ¢ ¡w c ¢¡n0−w k−c ¢ = 1 a − m + 1· ¡k c ¢¡n0 k ¢ ¡w c ¢¡n0−w k−c ¢ ≤ 1 a − m + 1( k s) s(1 + 1 k − s) k−s
by taking s = r = c in Lemma 2.3, and
ln[ µ w c ¶µ n0− k w − c ¶µ a − 1 a − m ¶ ] = ln[ µ w c ¶µ n0− k w − c ¶ ] + ln µ a − 1 a − m ¶ < k[1 + ln(n k + 1)] + ln µ a − 1 a − m ¶
by taking s = r = c in Lemma 2.4. Therefore, we have
ts(k, m, c, n; z) ≤ts(k, m, c, n0; z) ≤[(a − m + 1)(z − 1) + 1] ¡ a a−m+1 ¢¡n0 k ¢ ¡w c ¢¡n0−w k−c ¢¡a−1 a−m ¢ {1 + ln[ µ w c ¶µ n0− k w − c ¶µ a − 1 a − m ¶ ]} <(a − m + 1)(z − 1) + 1 a − m + 1 ( k s) s(1 + 1 k − s) k−s{1 + k[1 + ln(n k+ 1)] + ln µ a − 1 a − m ¶ } as required. Corollary 4.2.1 1. ts(k, m, n) = ts(k, m, 1, n; 1) < k k − m + 1(1 + 1 k − 1)k−1{1 + k[1 + ln( n k + 1)] + ln ¡k−1 k−m ¢ }. 2. ts(k, m, n; z) = ts(k, m, 1, n; z) < (k − m + 1)(z − 1) + 1 k − m + 1 (1 + 1 k − 1) k−1{1 + k[1 + ln(n k+ 1)] + ln ¡k−1 k−m ¢ }. 3. ts(k, m, c, n) = ts(k, m, c, n; 1) < 1 a − m + 1( k s) s(1 + 1 k − s) k−s{1 + k[1 + ln(n k + 1)] + ln ¡a−1 a−m ¢ }.
Moreover, we will give a better, but nonconstructive upper bound for the size of (k, m, n; z)-selectors in the following.
Lemma A (k, m, n; z)-selector is (m − 1, k − m + 1; z)-disjunct.
Proof. Suppose not. Then there exist k columns C1, C2, ..., Ck such that
| k [ i=m Ci\ m−1[ i=1 Ci| ≤ z − 1,
that is, there exist at most z − 1 rows with row weight 1 such that each of them hits exactly one of the columns Cm, Cm+1, ..., Ck. Hence there exist at most z − 1 disjoint m × k submatrices of the identity matrix Ik, it contradicts that M is a (k, m, n; z)-selector.
By the above lemma, an upper bound for the sizes the sizes of (k, m, n; z)-selectors can be obtained from that of (m − 1, k − m + 1; z)-disjunct matrices. Hence a nonconstructive upper bound (3.10, 36/42) of the sizes of (k, m, n; z)-selectors is given below. Put (d, r) = (m − 1, k − (m − 1)) in
t(n, d, r; z) as shown in Corollary 4.1.1, we have ts(k, m, n; z) < z k k − m + 1(1 + 1 k − 1) k−1{1 + k[1 + ln(n k + 1)] + ln µ k − 1 m − 1 ¶ }.
For the case z = 1, i.e.,
ts(k, m, n; 1) < k k − m + 1(1 + 1 k − 1) k−1{1 + k[1 + ln(n k+ 1)] + ln µ k − 1 m − 1 ¶ }
was given in Corollary 4.2.1 followed by a constructive proof in term of the Stein-Lov´asz theorem.
Some upper bounds for t(d, n), t(n, d, r], (n, d, r; z], t(n, d, r), t(n, d, r, s], ts(k, m, n) are survey in the following. 1. d-disjunct t(d, n) ≤ (d + 1)(1 +1 d) d{1 + ln[(d + 1)(¡ n d+1 ¢ −¡n−d−1 d+1 ¢ )]} [Y02] 2. 3. (d, r]-disjunct t(n, d, r] ≤ (1 +d r) r(1 + r d) d{1 + ln[¡n d ¢¡n−d r ¢ −¡n−(d+r)d ¢¡n−(d+r)−dr ¢]} [Y07] 4. (d, r; z]-disjunct t(n, d, r; z] < z(1 +d r) r(1r d) d{1 + k[1 + ln(n k + 1)]}, k = d + r [CFH08] 5. (d, r)-disjunct t(n, d, r) ≤ (1 +d r)(1 + r d) d{1 + ln[¡n d ¢¡n−d r ¢ −¡n−(d+r) d ¢¡n−(d+r)−d r ¢ ]} [Y07] 6. 7. (d, s out of r]-disjunct t(n, d, r, s] ≤ 1 + ln[ ¡n d ¢¡n−d r ¢ −¡n−(d+r)d ¢¡n−(d+r)−dr ¢] fd,r,s(p) for all 0 < p < 1, where fd,r,s(p) = (1 − p)d[1 − s−1 X i=0 µ r i ¶ pi(1 − p)r−i] [Y07]
8. 9. (k, m, n)-selector ts(k, m, n) ≤ ek 2 k − m + 1ln n k + ek(2k − 1) k − m + 1 [BGV05, by SLT] ts(k, m, n) ≤ ¡km m ¢ m![k(1 + 1 k − 1) k−1]{1 + ln[¡n k ¢ −¡n−k k ¢ ]} [Y07, by LLL] 10. 11. (k, m, w, n)-selector ts(k, m, w, n) ≤ [B08] 12.
4.2
Splitting systems, separating systems, covering designs and lotto
de-signs
An upper bound for SP (m, t) was given in [DSW04] by Lov´asz local lemma, and upper bounds for
SP (m, t), C(v, k, t), L(v, k, t, p) were given in [DZR08] by the Stein-Lov´asz theorem.
Upper bounds for SP (m, t; z), SE(m, t1, t2; z), C(v, k, t; z) and L(v, k, t, p; z) will be derived by using the extension of the Stein-Lov´asz theorem in the following theorems 4.3 ∼ 4.6.
Theorem 4.3 SP (m, t; z) ≤ z ¡m m 2 ¢ ¡t t 2 ¢¡m−t m 2−t2 ¢ {1 + ln[ µm 2 t 2 ¶2 ]}.
Proof. Let A be the binary matrix of order¡mt¢×¡mm
2
¢
with rows and columns indexed by {T | T ∈ ¡[m] t ¢ } and Γ = {B | B ∈¡[m] m 2 ¢
} respectively. The entry of A at the row indexed by the T and the
column indexed by the vector B ∈ Γ is 1 if B splits T ; and 0 otherwise. Observe that each row of A has weight v = ¡tt
2
¢¡m−t
m
2−t2
¢
, and each column of A has weight
a =¡m2 t 2 ¢¡m 2 t 2 ¢ =¡m2 t 2 ¢2
. By the extension of the Stein-Lov´asz Theorem, there exists a submatrix M of A of order [¡mt¢+¡mt 1 ¢¡m−t 1 t2 ¢
] × N with each row weight at least z, where
N ≤ z ¡m m 2 ¢ v {1 + ln a}.
It is straightforward to show that the columns of M form an uniform (m, t; z)-splitting system with
N blocks, as required.
In the following theorem we only discuss results where t16= t2. The cases when t1 = t2 can be handled in a similar way.
Theorem 4.4 SE(m, t1, t2; z) ≤ z¡mm 2 ¢ 2¡m−(t1+t2) m 2−t1 ¢ {1 + ln[2 µm 2 t1 ¶µm 2 t2 ¶ ]}.
Proof. Let A be the binary matrix of order [¡m t1 ¢¡m−t 1 t2 ¢ ] סm m 2 ¢
with rows and columns indexed by
{(T1, T2) | T1 ∈ ¡[m] t1 ¢ , T2 ∈ ¡[m] t2 ¢ with empty T1 T T2} and Γ = {B | B ∈ ¡[m] m 2 ¢ } respectively. The
entry of A at the row indexed by the pair (T1, T2) and the column indexed by the vector B ∈ Γ is 1 if B separates the pair (T1, T2); and 0 otherwise.
Observe that each row of A has weight v =¡m−(t1+t2) m 2−t1 ¢ +¡m−(t1+t2) m 2−t2 ¢ = 2¡m−(t1+t2) m 2−t1 ¢ , and each column of A has weight a =¡m2
t1 ¢¡m 2 t2 ¢ +¡m2 t2 ¢¡m 2 t1 ¢ = 2¡m2 t1 ¢¡m 2 t2 ¢
. By the extension of the Stein-Lov´asz Theorem, there exists a submatrix M of A of order [¡mt
1
¢¡m−t
1
t2
¢
] × N with each row weight at least
z, where N ≤ z ¡m m 2 ¢ v {1 + ln a}.
It is straightforward to show that the columns of M form an uniform (m, t1, t2; z)-splitting system with N blocks, as required.
Theorem 4.5 C(v, k, t; z) ≤ z ¡v t ¢ ¡k t ¢ {1+ ∈ µ k t ¶ }. Proof. Let A be the binary matrix of order¡v
t ¢
סv k ¢
with rows and columns indexed by {T | T ∈¡[v] t
¢
}
and Γ = {B | B ∈¡[v]k¢} respectively. The entry of A at the row indexed by the T and the column
indexed by the vector B ∈ Γ is 1 if T ⊆ B; and 0 otherwise. Observe that each row of A has weight ¡v−t
k−t ¢
, and each column of A has weight ¡k t ¢
. By the extension of the Stein-Lov´asz Theorem, there exists a submatrix M of A of order¡vt¢× N with each
row weight at least z, where
N ≤ z ¡v k ¢ ¡v−t k−t ¢ {1 + ln µ k t ¶ } = z ¡v t ¢ ¡k t ¢ {1 + ln µ k t ¶ }.
Note that the equality is obtained by counting the number of 1s in A in two ways. It is straightforward to show that the columns of M form an (v, k, t; z)-covering design with N blocks, as required.
Theorem 4.6 L(v, k, t, p; z) ≤ z µ v k ¶ min(t,k)X i=p µ t i ¶µ v − t k − i ¶ {1 + ln[ min(t,k)X i=p µ k i ¶µ v − k t − i ¶ ]}.
Proof. Let A be the binary matrix of order¡vt¢×¡vk¢with rows and columns indexed by {T | T ∈¡[v]t¢}
and Γ = {B | B ∈¡[v]k¢} respectively. The entry of A at the row indexed by the T and the column
indexed by the vector B ∈ Γ is 1 if |TTB| ≥ p; and 0 otherwise.
Observe that each row of A has weight min(t,k)X i=p µ t i ¶µ v − t k − i ¶
, and each of its column has weight min(t,k)X i=p µ k i ¶µ v − k t − i ¶
. By the extension of the Stein-Lov´asz Theorem, there exists a submatrix M of
A of order¡vt¢× N with each row weight at least z, where
N ≤ z µ v k ¶ min(t,k)X i=p µ t i ¶µ v − t k − i ¶ {1 + ln[ min(t,k)X i=p µ k i ¶µ v − k t − i ¶ ]}.
It is straightforward to show that the columns of M form an (v, k, t, p; z)-lotto design with N blocks, as required.
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