國立交通大學. Stein-Lovász 定理的推廣及其應用 An Extension of Stein-Lovász Theorem and Some of its Applications 研究生 : 李光祥 指導教授 : 黃大原教授 中華民國九十八年六月

Transcription

1 國立交通大學 應用數學系 碩士論文 Stei-Lovász 定理的推廣及其應用 A Extesio of Stei-Lovász Theorem a Some of its Applicatios 研究生 : 李光祥 指導教授 : 黃大原教授 中華民國九十八年六月

2 A Extesio of Stei-Lovász Theorem a Some of its Applicatios 研究生 : 李光祥 指導教授 : 黃大原 Stuet : Guag-Siag Lee Avisor : Tayua Huag 國立交通大學應用數學系碩士論文 A Thesis Submitte to Departmet of Applie Mathematics College of Sciece Natioal Chiao Tug Uiversity I Partial Fulfillmet of the Requiremets For the Degree of Master I Applie mathematics Jue 009 Hsichu, Taiwa, Republic of Chia 中華民國九十八年六月

3 誌 謝 這篇論文的完成, 首先要感謝指導教授黃大原教授, 從去年暑假開始就讓我進行論文的相關研究, 也常常提供一些學業上規劃的建議, 每當在課業或研究上有疑問的時候, 老師總是不厭其煩的替我解惑, 讓我得以順利的完成論文 同時要感謝傅恆霖教授 陳秋媛教授和翁志文教授的關心與幫 助, 使我在大學到研究所的這段時間學到了不少事情 感謝葉鴻國教授和傅東山教授擔任我的論文口試委員, 葉鴻國 教授提供我一些未來可以繼續研究的方向, 傅東山教授則是提 供我一些論文方面編寫的建議, 真的是非常感謝 還有要感謝 3 的夥伴 : 簡文昱 陳哲皓 林奕伸 黃瑞毅 李柏瑩 黃義閔以及陳鼎三, 謝謝他們陪伴我度過碩班這兩 年, 以及對我的照顧 最後要感謝我的父母, 因為有他們的支持, 讓我可以無憂無慮 的完成碩士學位

4 Stei-Lovász 定理的推廣及其應用 研究生 : 李光祥 指導教授 : 黃大原 國立交通大學 應用數學系 摘 要 Stei-Lovász 定理提供一個演算法的方式來找出好的覆蓋 coverig, 並且可以用來處理一些組合問題, 找出它們的上界 為了可以用來處理更多的組合問題, 在這篇論文裡, 我們將原先的 Stei-Lovász 定理作推廣 此外, 我們將利用推廣後的 Stei-Lovász 定理來處理一些模型, 這些模型包括 : 分離矩陣 isjuct matrices 選擇器selectors 以及系統集 set systems, 在固定行 colum 數的前提之下, 分別去找出這些矩陣的最小列 row 數的上界 其中分離矩陣和選擇器可以應用在匯集設計 poolig esig 上 i

5 A Extesio of Stei-Lovász Theorem a Some of its Applicatios Stuet : Guag-Siag Lee Avisor : Tayua Huag Departmet of Applie Mathematics Departmet of Applie Mathematics Natioal Chiao Tug Uiversity Natioal Chiao Tug Uiversity Hsichu, Taiwa Hsichu, Taiwa 30050

6 Abstract The Stei-Lovász theorem provies a algorithmic way to eal with the existece of goo coverigs a the to erive some upper bous relate to some combiatorial structures. I orer to eal with more combiatorial problems, a extesio of the classical Stei-Lovász theorem, calle the extee Stei-Lovász theorem, will be give i this thesis. Moreover, we will also iscuss applicatios of the extee Stei-Lovász theorem to various moels state as follows:. Several isjuct matrices for group testig purpose -isjuct matrices, ; z]-isjuct matrices;, r]-isjuct matrices,, r; z]-isjuct matrices;, r-isjuct matrices,, r; z-isjuct matrices;, s out of r]-isjuct matrices,, s out of r; z]-isjuct matrices.. Several selectors for group testig purpose, m, -selectors,, m, ; z-selectors;, m, c, -selectors,, m, c, ; z-selectors. 3. Some set systems for others uiform m, t-splittig systems, uiform m, t; z-splittig systems; uiform m, t, t -separatig systems, uiform m, t, t ; z-separatig systems; v,, t-coverig esigs, v,, t; z-coverig esigs; v,, t, p-lotto esigs, v,, t, p; z-lotto esigs. iii

7 Cotets 摘要 Abstract i iii Itrouctio Prelimiaries 7. Several isjuct matrices Several selectors Some set systems Some basic coutig results The Stei-Lovász Theorem a its extesio 0 3. The Stei-Lovász Theorem Extesio of The Stei-Lovász Theorem Some Applicatios of the extee Stei-Lovász Theorem 8 4. Bous for several isjuct matrices Bous for several selectors Bous for some set systems Coclusio 59 Appeix iv

8 Bibliography 6 v

9 Chapter Itrouctio Let X be a fiite set a let Γ be a family of subsets of X. We eote by H = X, Γ the hypergraph havig X as the set of vertices a Γ as the set of hypereges. The egree of x X is the umber of hypereges cotaiig x. Deote by H the maximum egree i the hypergraph H. For H a other fuctios to be efie we remove the argumet H if o cofusio ca arise. A biary matrix M = m ij of orer Γ X ca be iterprete as a bloc-poit iciece matrix of the hypergraph H, i.e., the rows of M correspo to the hyperege set {E, E,..., E Γ }, a the colums correspo to the vertex set {x, x,..., x X }, where { if the hyperege Ei cotais the vertex x m ij = j 0 otherwise. The weight of a biary matrix M is the umber of etries with a. A subset M Γ the same hyperege may occur more tha oce such that each vertex belogs to at most of its members is calle a -matchig of the hypergraph H. The maximum size over all -matchigs of the hypergraph H is eote by ν H. A -matchig is simple if o hyperege occurs i it more tha oce. Deote by ν the maximum umber of hypereges i simple -matchigs, the ν ν. A subset T X i this thesis, the same vertex oes ot occur more tha oce such that T E for ay hyperege E is calle a -cover of the hypergarph H. The miimum size over all -covers of the hypergraph H is eote by τ H. Thus τh = τ H is the

10 miimum size of a vertex cover of the hypergraph H. A vector w E, w E,..., w E Γ with w Ei 0 for each E i Γ is calle a fractioal matchig of the hypergraph H if each etry of the vector w E, w E,..., w E Γ M is at most. A vector w x, w x,..., w x X with w xi 0 for each x i X is calle a fractioal cover of the hypergraph H if each etry of the vector Mw x, w x,..., w x X t is at least. Defie ν H = max E i Γ w Ei a τ H = mi x i X w xi, where the extrema are tae over all fractioal matchigs w E, w E,..., w E Γ a all fractioal covers w x, w x,..., w x X, respectively. By the uality theorem of liear programmig, we have ν = τ. The it is easy to see that ν ν / ν = τ τ /. Oe of the most atural methos to prouce a small vertex cover of a give hypergraph H is the so-calle Greey Cover Algorithm, which we escribe as follows:. Let x be a vertex with maximum egree.. Suppose that x, x,..., x i have bee alreay selecte, if x, x,..., x i cover all hypereges, the we stop; otherwise, let x i+ be a vertex which covers the most umber of ucovere hypereges. Geerally, the greey cover algorithm is ot the best, but we ca expect that it gives a rather goo estimate. By the greey cover algorithm, a upper bou for τh was give by Lovász [0]. Theorem. [0] If H is a hypergraph a ay greey cover algorithm prouces t coverig vertices, the t ν + ν ν + ν. Use the facts that τ t a ν i ν i iν = iτ, we have the followig corollary.

11 Corollary. [0] For a hypergraph H, τh τ H < + l τ H. Hece we have the followig theorem for completeess, we also give a proof. Theorem. For a hypergraph H = X, Γ, τh < where = max x X {E : E Γ with x E}. X + l, mi E Γ E Proof. Let M be the bloc-poit iciece matrix of H. Defie w xi = mi E Γ E for each x i X. The each E i -etry of the vector Mw x, w x,..., w x X t is E i mi E Γ E, i.e., w x, w x,..., w x X is a fractioal cover of H. Hece τ H x i X w xi = X mi E Γ E. By Corollary, τh < + l τ H X + l, mi E Γ E as require. Similarly, by the greey cover algorithm, a equivalet statemet i terms of the poitbloc iciece matrices of the correspoig hypergraphs was give by Stei [] iepeetly. 3

12 Theorem 3. [] Let X be a fiite set of cariality, a let Γ = {A, A,..., A t } be a family of subsets of X, where A i a for all i t. Assume that each elemet of X is i at least q members of the set Γ. The there is a subfamily of Γ that covers X a has at most a + t q a members. Note that Theorem 3 is closely relate to wor of Fulerso a Ryser [9] i the -with of a 0, -matrix. They efie the -with of such a matrix, A, as the miimum umber of colums that ca be selecte from A i such a way that each row of the resultig submatrix has at least oe. I this termiology, Theorem 3 ca be restate as follows: Theorem 4. [] Let A be a 0, -matrix with rows a t colums. Assume that each row cotais at least q s a each colum at most a s. The the -with of A is at most a + t q a. Theorem 4 was calle the Stei-Lovász Theorem i [6] while ealig with the coverig problems i coig theory. The Stei-Lovász theorem was first use i ealig with the upper bous for the sizes of, m, -selectors []. Ispire by this wor, it was also use i ealig with the upper bous for the sizes of, r; z]-isjuct matrices [5]. Some more applicatios ca also be fou i [8]. The otio of, m, -selectors was first itrouce by De Bois, Gasieiec a Vaccaro i [], followe by a geeralizatio to the otio of, m, c, -selectors []. A further geeralizatio of, m, c, -selectors will be give i Chapter. I this thesis, efiitios of several properties over biary matrices are cosiere i Chapter icluig several isjuct matrices, several selectors a some set systems state as follows: 4

13 . Several isjuct matrices -isjuct matrices, ; z]-isjuct matrices;, r]-isjuct matrices,, r; z]-isjuct matrices;, r-isjuct matrices,, r; z-isjuct matrices;, s out of r]-isjuct matrices,, s out of r; z]-isjuct matrices.. Several selectors, m, -selectors,, m, ; z-selectors;, m, c, -selectors,, m, c, ; z-selectors. 3. Some set systems uiform m, t-splittig systems, uiform m, t; z-splittig systems; uiform m, t, t -separatig systems, uiform m, t, t ; z-separatig systems; v,, t-coverig esigs, v,, t; z-coverig esigs; v,, t, p-lotto esigs, v,, t, p; z-lotto esigs. Note that the upper bous of the sizes of several isjuct matrices a selectors are obtaie for group testig purpose, a the upper bous of the sizes of some set systems are obtaie for others. Some formulas are give i Sectio.4 for later simplificatio purpose use i Chapter 4. I orer to eal with the upper bous for these biary matrices efie i Chapter, a extee Stei-Lovász theorem is erive i Chapter 3. Some applicatios of the etermiatio of some upper bous of the sizes of various moels are cosiere i Chapter 4. I Sectio 4. a Sectio 4., the extee Stei-Lovász theorem will be use i ealig the upper bous for the sizes of several isjuct matrices a selectors, respectively. Those 5

14 upper bous for the sizes of uiform splittig systems, uiform separatig systems, coverig esigs a lotto esigs are give i Sectio 4.3 respectively. 6

15 Chapter Prelimiaries. Several isjuct matrices A few types of biary matrices, calle isjuct matrices, will be itrouce i this sectio, followe by correspoig associate parameters. These families of isjuct matrices will be use as moels for poolig esigs. Defiitio... A biary matrix M of orer t is calle -isjuct if the uio of ay colums oes ot cotai ay other colum of M, i.e., for ay + colums C, C,, C +, C + \ C i. The iteger t is calle the size of the -isjuct matrix. The miimum i= size over all -isjuct matrices with colums is eote by t,. Defiitio... A biary matrix M of orer t is calle ; z]-isjuct if for ay + colums C, C,, C +, C + \ C i z. The iteger t is calle the size of the ; z]- i= isjuct matrix. The miimum size over all ; z]-isjuct matrices with colums is eote by t, ; z]. Defiitio..3. A biary matrix M of orer t is calle, r]-isjuct if the uio of ay colums oes ot cotai the itersectio of ay other r colums of M, i.e., for ay r +r + r colums C, C,, C +r, C i \ C i. The iteger t is calle the size of the i= i=r+, r]-isjuct matrix. The miimum size over all, r]-isjuct matrices with colums is eote by t,, r]. 7

16 Defiitio..4. A biary matrix M of orer t is calle, r; z]-isjuct if for ay r +r + r colums C, C,, C +r, C i \ C i z. The iteger t is calle the size of the i= i=r+, r; z]-isjuct matrix. The miimum size over all, r; z]-isjuct matrices with colums is eote by t,, r; z]. Defiitio..5. A biary matrix M of orer t is calle, r-isjuct if the uio of ay colums oes ot cotai the uio of ay other r colums of M, i.e., for ay r +r + r colums C, C,, C +r, C i \ C i. The iteger t is calle the size of the i= i=r+, r-isjuct matrix. The miimum size over all, r-isjuct matrices with colums is eote by t,, r. Defiitio..6. A biary matrix M of orer t is calle, r; z-isjuct if for ay r +r + r colums C, C,, C +r, C i \ C i z. The iteger t is calle the size of i= i=r+ the, r; z-isjuct matrix. The miimum size over all, r; z-isjuct matrices with colums is eote by t,, r; z. Defiitio..7. A biary matrix M of orer t is calle, s out of r]-isjuct, s r, if for ay colums a ay other r colums of M, there exists a row iex i which oe of the colums appear a at least s of the r colums o. The iteger t is calle the size of the, s out of r]-isjuct matrix. The miimum size over all, s out of r]-isjuct matrices with colums is eote by t,, r, s]. Defiitio..8. A biary matrix M of orer t is calle, s out of r; z]-isjuct, s r, if for ay colums a ay other r colums of M, there exist z row iices i which oe of the colums appear a at least s of the r colums o. The iteger t is calle the size of the, s out of r; z]-isjuct matrix. The miimum size over all, s out of r; z]-isjuct matrices with colums is eote by t,, r, s; z]. 8

17 Some subclasses of, s out of r; z]-isjuct matrices are liste i the followig table. parameters types bous refereces s = r =, z = -isjuct t, [3] s = r = ; z]-isjuct s = r, z =, r]-isjuct t,, r] [4] s = r, r; z]-isjuct t,, r; z] [5] s =, z =, r-isjuct t,, r [4] s =, r; z-isjuct t,, r; z z =, s out of r]-isjuct t,, r, s] [4], s out of r; z]-isjuct t,, r, s; z]. Several selectors A few types of biary matrices, calle selectors, will be itrouce i this sectio, followe by correspoig associate parameters. These families of selectors will be use as moels for poolig esigs. Defiitio... For itegers, m a with m, a biary matrix M of orer t is calle a, m, -selector if ay t submatrix of M cotais a submatrix with each row weight exactly oe, with at least m istict rows. The iteger t is calle the size of the, m, -selector. The miimum size over all, m, -selectors is eote by t s, m,. Defiitio... For itegers, m a with m, a biary matrix M of orer t is calle a, m, ; z-selector if ay t submatrix of M cotais z isjoit submatrices with each row weight exactly oe, with at least m istict rows each. The iteger t is calle the size of the, m, ; z-selector. The miimum size over all, m, ; z-selectors is eote by t s, m, ; z. Defiitio..3. For itegers, m, c a with c a m c, a t biary matrix M is calle a, m, c, -selector if ay t submatrix of M cotais a submatrix with each row weight exactly c, with at least m istict rows. The iteger t is 9

18 calle the size of the, m, c, -selector. The miimum size over all, m, c, -selectors is eote by t s, m, c,. Defiitio..4. For itegers, m, c a with c a m c, a t biary matrix M is calle a, m, c, ; z-selector if ay t submatrix of M cotais z isjoit submatrices with each row weight exactly c, with at least m istict rows each. The iteger t is calle the size of the, m, c, ; z-selector. The miimum size over all, m, c, ; z-selectors is eote by t s, m, c, ; z. It is iterestig to remar that the otio of, m, -selectors was first itrouce by De Bois, Gasieiec a Vaccaro [], a it was the geeralize to the otio of, m, c, - selectors [], which are equivalet to, m,, ; -selectors a, m, c, ; -selectors rsepectively. The upper bous for the sizes of, m, -selectors a, m, c, -selectors were stuie i [] a i [] respectively by the Stei-Lovász theorem. The bous for the sizes of, m, c, ; z-selectors will be erive by the extee Stei-Lovász theorem Theorem 3.. i Chapter 4 Theorem 4... Some subclasses of, m, c, ; z-selectors are liste i the followig table. parameters types bous refereces c =, z =, m, -selectors t s, m, [, 4] c =, m, ; z-selectors t s, m, ; z z =, m, c, -selectors t s, m, c, [], m, c, ; z-selectors t s, m, c, ; z The relatioship betwee various moels isjuct matrices, selectors a oaaptive group testig are liste below.. A, r]-isjuct matrix ca be use to ietify the up-to- positives o the complex moel [4].. The property of h, -isjuctess is a ecessary coitio for ietifyig the positive set o the, h-ihibitor moel [3]. 0

19 3. There exists a two-state group testig algorithm for fiig up-to- positives out of items a that uses a umber of tests equal to t +, where t is the size of a, +, -selector []..3 Some set systems Most of the combiatorial structures ca be viewe as set systems. We preset some relevat efiitios. A set system is a pair X, Γ, where X is a set of poits a Γ is a set of subsets of X, calle blocs. A set sysyem X, Γ is calle -uiform if B = for each B Γ. Defiitio.3.. Let m a t be eve itegers with t m. A uiform m, t-splittig system is a pair X, Γ where X = m, Γ is a family of m subsets of X, calle blocs such that for every T X with T = t, there exists a bloc B Γ such that T B = t, i.e., B splits T. The system X, Γ is also calle a t-splittig system. The miimum umber of blocs over all t-splittig systems is eote by SP m, t. Defiitio.3.. Let m a t be eve itegers with t m, a let z be a positive iteger. A uiform m, t; z-splittig system is a pair X, Γ where X = m, Γ is a family of m subsets of X, calle blocs such that for every T X with T = t, there exist z blocs B Γ such that T B = t, i.e., B splits T. The system X, Γ is also calle a t; z- splittig system. The miimum umber of blocs over all t; z-splittig systems is eote by SP m, t; z. Defiitio.3.3. Let m be a eve iteger, a let t, t be positive itegers with t +t m. A uiform m, t, t -separatig system is a pair X, Γ where X = m, Γ is a family of m subsets of X, calle blocs such that for every T, T X, where T i = t i for i =, a T T =, there exists a bloc B Γ for which either T B, T B = or T B, T B =, i.e., T, T are separate by B. The system X, Γ is also calle a

20 t, t -separatig system. The miimum umber of blocs over all t, t -separatig systems is eote by SEm, t, t. Defiitio.3.4. Let m be a eve iteger, a let t, t, z be positive itegers with t +t m. A uiform m, t, t ; z-separatig system is a pair X, Γ where X = m, Γ is a family of m subsets of X, calle blocs such that for every T, T X, where T i = t i for i =, a T T =, there exist z blocs B Γ for which either T B, T B = or T B, T B =, i.e., T, T are separate by B. The system X, Γ is also calle a t, t ; z- separatig system. The miimum umber of blocs over all t, t ; z-separatig systems is eote by SEm, t, t ; z. Defiitio.3.5. Let v,, a t be positive itegers with t v. A v,, t-coverig esig is a pair X, Γ where X = v, Γ is a family of -subsets of X, calle blocs such that for every T X with T = t, there exists a bloc B Γ cotaiig T. The miimum umber of blocs over all v,, t-coverig esigs is eote by Cv,, t. Defiitio.3.6. Let v,, t a z be positive itegers with t v. A v,, t; z-coverig esig is a pair X, Γ where X = v, Γ is a family of -subsets of X, calle blocs such that for every T X with T = t, there exist z blocs B Γ cotaiig T. The miimum umber of blocs over all v,, t; z-coverig esigs is eote by Cv,, t; z. Defiitio.3.7. Let v,, t, a p be positive itegers with p t, v. A v,, t, p-lotto esig is a pair X, Γ where X = v, Γ is a family of -subsets of X, calle blocs such that for every T X with T = t, there exists a bloc B Γ such that T B p. The miimum umber of blocs over all v,, t, p-lotto esigs is eote by Lv,, t, p. Defiitio.3.8. Let v,, t, p a z be positive itegers with p t, v. A v,, t, p; z- lotto esig is a pair X, Γ where X = v, Γ is a family of -subsets of X, calle blocs such that for every T X with T = t, there exist z blocs B Γ such that T B p. The miimum umber of blocs over all v,, t, p; z-lotto esigs is eote by Lv,, t, p; z.

21 Note that whe p = t, a v,, t, t; z-lotto esig will be reuce to a v,, t; z-coverig esig. The relate bous are summarize i the followig table. types bous refereces uiform m, t-splittig systems SP m, t [8] uiform m, t; z-splittig systems SP m, t; z uiform m, t, t -separatig systems SEm, t, t uiform m, t, t ; z-separatig systems SEm, t, t ; z v,, t-coverig esigs Cv,, t [8] v,, t; z-coverig esigs Cv,, t; z v,, t, p-lotto esigs Lv,, t, p [8] v,, t, p; z-lotto esigs Lv,, t, p; z.4 Some basic coutig results Stei-Lovász theorem a its extesio will be use to estimate the upper bous of the sizes for poolig esigs of various moels. I orer to give upper bous for the above metioe parameters, the followig results ivolvig biomial coefficiets will be ivolve. Lemma.4.3 will be use i showig appropriate values of w for poolig esigs of various moels. We ee iformatio regarig the maximum of the fuctio w w w fw = r s s with various r a s whe ealig with possible upper bous for t of various moels. Lemmas will be use i the simplificatios of the bous M, a l a respectively i the v expressio M + l a fou i the Stei-Lovász theorem Theorem 3... v Lemma.4.. The fuctio fx = + l x x is strictly ecreasig o,. 3

22 Proof. f x = x + l x x x = l x x < 0 for all x,, as require. Lemma.4.. a ab b b! ea b b Proof. Sice e x = 0 Therefore, as require. a = b x!, we have ex xb b! a! a b!b! = aa a b + b! for each x, thus e b bb b! ab b! ab b! b!eb b b = ea b b, a hece b!eb b b. Lemma.4.3. For ay positive itegers,, r, s with = + r a s r, the fuctio w w w fw = r s s gets its maximum at w = s s. Proof. First we ote that w w w fw = r s s w! = w!! w! w w r + s!r s! + r s! s + r s! w w + r s w w s = =. + r s s s s 4

23 Sice w + w + s fw + = s s w s w w + w s = w s w + s s w s w + = w w + s fw, a w s w w + w + s = w + w w + s w + w s w = s s if a oly if s w = w + s, i.e., w = ; hece w s w + s s for w a w w + s w s w w + w + s s s for w. As a cosequece, we the have s s fw is icreasig for w, a s s fw is ecreasig for w as require. w w w By taig s = r = c i fw =, we get the quaratic fuctio r s s w w gw = for selectors. Hece we have the followig corollary. c c Corollary.4.. The fuctio w w gw = c c gets its maximum at w = c c. 5

24 Lemma.4.4. For ay positive itegers,, r, s with = + r a s r, s + s s r s s s s Proof. Without loss of geerality, let s s a thus ote that the left ha sie is = s + s s r s s s s i 0 i s 0 i r+ i s 0 j s j. s. Moreover, we s s To prove this iequality, we will rearrage the terms i the eomiator so that for each t with 0 t r +, i.e., we will give a bijectio t ft f : {0,,..., r + } {i s 0 i s } {j s 0 j s } with the property that ft t for each t. Note that the elemet 0 will be coute twice as 0 a 0 respectively i the rage of the fuctio f. Note also that if i s s s the s < s = i+i s s > t, i a hece j = j+ s < j+ i = j t = j t = t, where i+j = t. t i s s t i t i j A such fuctio f is efie recursively as follows. For t = 0,,, let f0 = 0, f = s 0, f =. For 3 t r +, let i resp. j be the smallest positive itegers such s s that i resp. j / {f0, f,..., ft } if they exist, it follows that t = i + j. s s. Let ft = i s if i s t.. Otherwise, we efie ft = j s. 3. Fially, suppose t is large a there is o such i, ote that s we have s r + a r + <, + + s t t s s s s for all s t r +, we efie ft = t s s. 6

25 Cearly, the fuctio f efie above is -, oto, a ft t for all 0 t r + as require. Lemma.4.5. For ay positive itegers,, r, s with = + r a s r, let be the smallest positive iteger such that w = s is a iteger, the r w s s s s. w w r r s s s Proof. = = r w w w r s s w! w!!!!!! r!r! w! w r + s!r s! w! w s!s! r! s + s s r + ww w s + r s!s! w w w r + s + = r s s + s s r + s s s s + s s s r + s + = r s s s s + s s r + s s s + s s + s s s s s by Lemma.4.4. r s Lemma.4.6. For ay positive itegers,, r, s with = + r a s r, let be the smallest positive iteger such that w = s is a iteger, the w w w l < [ + l s r s s + ] + l. 7

26 Proof. First we ote that < + for such. Sice Therefore, w w w r s s = w! w!! w! w r + s!r s! w w + r s = + r s s w w s = s s e s s e s s s s s =e s e s s =e s <e + s s =e +. w w l r s < le s + w s a ea b b b, we have w + r s! s + r s! = [ + l + ] + l s. 8

27 The substitutios of w for various subclasses are summarize i the followig table: types parameters -isjuct s = r =, z = w = ; z]-isjuct s = r = w =, r]-isjuct s = r, z = w = r, r; z]-isjuct s = r w = r, r-isjuct s =, z = w =, r; z-isjuct s = w =, s out of r]-isjuct z = s s w =, s out of r; z]-isjuct s s w = w = w = w = r w = r w = w = w = s w = s types parameters, m, -selectors c =, z = w =, m, ; z-selectors c = w =, m, c, -selectors z = w =, m, c, ; z-selectors w = c c c c w = w = w = c w = c 9

28 Chapter 3 The Stei-Lovász Theorem a its extesio 3. The Stei-Lovász Theorem We ow itrouce the Stei-Lovász theorem as follows. The Stei-Lovász theorem was first use by Stei [] a Lovász [0] i stuyig some combiatorial coverig problems. I [6], the authors applie this theorem to some problems i coig theory. The Stei- Lovász theorem is ow state a the proof is iclue for completeess [8], with a mior moificatio. Theorem 3... [8] Let A be a 0, matrix with N rows a M colums. Assume that each row cotais at least v oes, a each colum at most a oes. The there exists a N K submatrix C with K < N a + M v l a M v + l a, such that C oes ot cotai a all-zero row. Proof. A costructive approach for proucig C is presete. Let A a = A. We begi by picig the maximal umber K a of colums from A a, whose supports are pairwise isjoit a each colum havig a oes perhaps, K a = 0. Discarig these colums a all rows iciet to oe of them, we are left with a a M K a matrix A a, where a = N ak a. Clearly, 0

29 the colums of A a have at most a oes iee, otherwise such a colum coul be ae to the previously iscare set, cotraictig its maximality. Now we remove from A a a maximal umber K a of colums havig a oes a whose supports are pairwise isjoit, thus gettig a a M K a K a matrix A a, where a = N ak a a K a. The process will termiate after at most a steps. The uio of the colums of the iscare sets form the esire submatrix C with The first step of the algorithm gives K = a K i. i= a = N ak a, which we rewrite, settig a+ = N, as Aalogously, K a = a+ a. a K i = i+ i, i =,..., a. i Now we erive a upper bou for i by coutig the umber of oes i A i i two ways: every row of A i cotais at least v oes, a every colum at most i oes, thus v i i M K a K i i M. Furthermore, K = a K i = i= a i= i+ i i = a+ a + a aa + a a a + + N/a + M/v/a + /a + + /, thus givig the result.

30 The greey proceure as show i the proof costructs the esire submatrix oe colum at a time, a hece the algorithm below follows [8]. Algorithm: STEIN-LOVÁSZA iput: A is a N M matrix, each colum has at most a oes, each row has at least v oes C while A has at least oe row o fi a colum c i A havig maximum weight elete all rows of A that cotai a i colum c elete colum c from A output: Returs a submatrix of A with o all-zero row At each state, a ew colum is ae to the submatrix that maximizes the umber of ew rows that are yet ucovere. Whe all rows are covere, the algorithm stops. It seems quite iterestig that we ca use the Stei-Lovász theorem to erive bous for some combiatorial array [8]. 3. Extesio of The Stei-Lovász Theorem The Stei-Lovász theorem ca be further extee from rows of the resultig submatrix with weight at least to the case of rows of the resultig submatrix with weight at least z. The bou ca be further improve whe A is a matrix with costat row weight a colum weight as well, i.e., i the laguage of hypergraphs, uiform a regular.

31 Theorem 3... Let A be a 0, matrix of orer N M, a let v, a, z be positive itegers. Assume that each row cotais at least v oes, a each colum at most a oes. The there exists a submatrix C of orer N K with K < v v z zm v such that each row of C has weight at least z. + l a, More specifically, if the matrix is v-uiform a a-regular, the upper bou ca the be reuce to K < z M v + l a. The strategy for the proof of Theorem 3.. is as follows:. Use the Stei-Lovász theorem to obtai a submatrix C with each row has weight at least.. Choose some colums i the matrix A\C to combie with the submatrix C to form a submatrix C with each row has weight at least. 3. Choose some colums i the matrix A\C to combie with the submatrix C to form a submatrix C 3 with each row has weight at least Step by step, a fially we obtai the esire submatrix C = C z with each row has weight at least z. Note that this upper bou maes sese oly if v v z zm v + l a < M, i.e., z < v + + l a 3

32 i geeral, or if z M v + l a < M, i.e., z < v + l a for the case of uiform a regular. Proof. A costructive approach for proucig C is presete. Let A = A. By the Stei- Lovász theorem, there exists a N M submatrix C = B = B of A with M < M v + l a such that each row of C has weight at least. The algorithm use i the proof of the Stei-Lovász theorem shows that some rows of C have weight exactly. Let R be the set of iices of those rows a R = r. Let A be the submatrix of orer r M M obtaie from A by eletig the submatrix C a the i-th row, i / R as well. The each row of A cotais at least v oes, a each colum at most a oes. Agai, by the Stei-Lovász theorem, there exists a r M submatrix B with M < M M + l a such that each row of v B has weight at least. Let B be the matrix of orer N M obtaie from B by aig the i-th row, i / R. Let C be the matrix of orer N M + M obtaie by the uio of B a B. The C is a submatrix of A with each row weight at least. Similarly, there exist some rows of C that have weight exactly. Let R be the set of iices of those rows a R = r. Cotiue i this way, we have: For i z, i. A i is a matrix of orer r i M M j, a each row cotais at least v i j= oes, a each colum at most a oes.. B i is a r i M i submatrix of A i with M i < 4 i M j= M j v i + l a, a each row has

33 weight at least. For i z, 3. B i is a matrix of orer N M i. 4. C i is a N i M j submatrix of A, a each row has weight at least i. j= Hece, C = C z is the submatrix require, that is, K = z M j = M + M + + M z j= < M v + l a + M M + l a + + v < M v thus gives the result. M M + l a + + l a + + v = M + l a v + v + + v z z M j= M j + l a v z + l a v z < M + l a v z + v z + + v z z = M + l a v z v = v z zm + l a, v More specifically, for the case of uiform a regular, usig similar argumet as above with a mior moificatio. First we ote that Nv = Ma by coutig the weight of A i two i ways. For i z, A i is a matrix of orer r i M M j, a each row cotais exactly v i oes, a each colum at most a oes. Moreover, a lower bou for i M j is erive by coutig the weight of the submatrix C i i two ways; each row of C i j= i cotais at least i oes, a each colum exactly a oes, thus Ni M j a, a hece M i v i M j for i z. Furthermore, j= j= j= 5

34 K = z M j = M + M + + M z j= z M < M v + l a + M M + l a l a v v z M M v + l a + M M M z v v + l a + + v + l a v z j= M j = M v + l a + M v + l a + + M v + l a = z M v + l a, thus gives the result. Remar 3... Sice this upper bou maes sese oly if z < v + + l a i geeral, z < v + + l a < v + < v + = v +. The z < v a thus v v z = + z v z < + =. Hece K < v v z z M v + l a < z M v + l a for geeral case. 6

35 Similarly, Theorem 3.. ca be restate i the laguage of hypergraphs i the followig corollary. Recall that a subset T X such that T E z for ay hyperege E is calle a z-cover of the hypergarph H, a the miimum size of a z-cover of the hypergraph H is eote by τ z H. Corollary 3... For a hypergraph H = X, Γ a a positive iteger z, τ z H < where = max x X {E : E Γ with x E}. z X + l, mi E Γ E More specifically, for the case of uiform a regular, we have the followig corollary. Corollary 3... Let H = X, Γ be a v-uiform a a-regular hypergraph with vertex set X a ege set Γ, the τ z H < z X + l a. v We cojecture that τ z H zτ H hols for hypergraphs which are uiform a regular. However, it is ot true i geeral as show i the followig example. For the hypergraph H with X = {,, 3,..., 8} a Γ = {{,, 3}, {4, 5, 6}, {, 7, 8}}. It is easy to see that {, 4} is a -cover with miimum size, hece τ H =. Similarly, {,, 4, 5, 7} is a -cover with miimum size, hece τ H = 5. This shows that τ H = 5 > = τ H. 7

36 Chapter 4 Some Applicatios of the extee Stei-Lovász Theorem The Stei-Lovász theorem was first use i ealig with the upper bous for the sizes i the moel of, m, -selecters []. Ispire by this wor, it was also use i ealig with the upper bous for the sizes of, r; z]-isjuct matrices [5]. I Sectio 4. a Sectio 4., the extee Stei-Lovász theorem will be use i ealig the upper bous for the sizes of several isjuct matrices Theorem a selectors Theorem , respectively. Those upper bous for the sizes of uiform splittig systems, uiform separatig systems, coverig esigs a lotto esigs are give i Sectio 4.3 Theorem respectively. 4. Bous for several isjuct matrices Note that upper bous for the sizes of -isjuct matrices,, r]-isjuct matrices,, r- isjuct matrices a, s out of r]-isjuct matrices were give i [3, 4] by the Lovász Local Lemma. Recall that t, is the miimum size over all -isjuct matrices with colums. Theorem 4... For ay positive itegers a, if = +, the t, < { + [ + l + ]}. 8

37 Proof. For w, let A be the biary matrix of orer [ ] w [] with rows a colums iexe by {D, s D, s [] \ D} a V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, s a the colum iexe by the vector v V is if the etries of v over D are all zero a the etry of v at s is oe; a 0 otherwise. + Observe that each row of A has weight, a each colum of A has weight w w w. By the Stei-Lovász theorem, there exists a submatrix M of A of orer [ ] t havig o zero rows, where t < w w { + l[ + w w ]} = w { + l[ w w w ]}. Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a -isjuct matrix of orer t. We the have t, < w { + l[ w w w ]}. Let be the smallest positive iteger such that w = is a iteger. We have by Lemma.4.5 taig s = r =, a w l w w w < [ + l + ] 9

38 by Lemma.4.6 taig s = r =. Therefore, t, t, < { + l[ w w w < { + [ + l + ]} w ]} as require. Recall that t, ; z] is the miimum size over all ; z]-isjuct matrices with colums. Theorem 4... For ay positive itegers, a z, if = +, the t, ; z] < z { + [ + l + ]}. Proof. For w, let A be the biary matrix of orer [ ] w [] with rows a colums iexe by {D, s D, s [] \ D} a V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, s a the colum iexe by the vector v V is if the etries of v over D are all zero a the etry of v at s is oe; a 0 otherwise. + Observe that each row of A has weight, a each colum of A has weight w w w. By the extee Stei-Lovász theorem, there exists a submatrix M of A of orer [ ] t with each row weight at least z, where t < z w w { + l[ + w w ]} = z w { + l[ w w w Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a ; z]-isjuct matrix of orer t. We the ]}. 30

39 have t, ; z] < z w { + l[ w w w ]}. Let be the smallest positive iteger such that w = is a iteger. We have by Lemma.4.5 taig s = r =, a w l w w w by Lemma.4.6 taig s = r =. Therefore, t, ; z] t, ; z] < < [ + l + ] z w { + l[ w w < z { + [ + l + ]} w ]} as require. Recall that t,, r] is the miimum size over all, r]-isjuct matrices with colums. Theorem For ay positive itegers, a r, if = + r, the t,, r] < r r { + [ + l + ]}. Proof. For r w, let A be the biary matrix of orer [ ] with r w [] [] rows a colums iexe by {D, R D, R with D R empty} a r 3

40 V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, R a the colum iexe by the vector v V is if the etries of v over D are all zero a the etries of v over R are all oe; a 0 otherwise. + r Observe that each row of A has weight, a each colum of A has weight w r w w. By the Stei-Lovász theorem, there exists a submatrix M of A of orer r [ ] t havig o zero rows, where r t < w w { + l[ + r w r w r ]} = r w { + l[ w w r w r ]}. Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a, r]-isjuct matrix of orer t. We the have t,, r] < r w { + l[ w w r w r ]}. Let be the smallest positive iteger such that w = r by Lemma.4.5 taig s = r, a r w w r r r w l w r < [ + l + ] is a iteger. We have 3

41 by Lemma.4.6 taig s = r. Therefore, as require. t,, r] t,, r] < r { + l[ w w r w < r r { + [ + l + ]} w ]} r Recall that t,, r; z] is the miimum size over all, r; z]-isjuct matrices with colums. Theorem [5] For ay positive itegers,, r a z, if = + r, the t,, r; z] < z r r { + [ + l + ]}. Proof. For r w, let A be the biary matrix of orer [ ] with r w [] [] rows a colums iexe by {D, R D, R with D R empty} a r V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, R a the colum iexe by the vector v V is if the etries of v over D are all zero a the etries of v over R are all oe; a 0 otherwise. + r Observe that each row of A has weight, a each colum of A has weight w r w w. By the extee Stei-Lovász theorem, there exists a submatrix M of A of r orer [ ] t with each row weight at least z, where r t < z w w { + l[ + r w r w r ]} = z r w { + l[ w w r w r ]}. Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a, r; z]-isjuct matrix of orer t. We 33

42 the have t,, r] < z r w { + l[ w w r w r ]}. Let be the smallest positive iteger such that w = r by Lemma.4.5 taig s = r, a r w w r r r w l w r by Lemma.4.6 taig s = r. Therefore, t,, r; z] t,, r; z] < < [ + l + ] z r w { + l[ w w r < z r r { + [ + l + ]} is a iteger. We have w r ]} as require. Recall that t,, r is the miimum size over all, r-isjuct matrices with colums. Theorem For ay positive itegers, a r, if = + r, the t,, r < r + { + [ + l + ] + l }. Proof. For w, let A be the biary matrix of orer [ ] with r w [] [] rows a colums iexe by {D, R D, R with D R empty} a r 34

43 V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, R a the colum iexe by the vector v V is if the etries of v over D are all zero a at least oe etry of v over R is oe; a 0 otherwise. mir,w r + r Observe that each row of A has weight, a each colum of A j w j j= mir,w w w w has weight. By the Stei-Lovász theorem, there exists a r j j j= submatrix M of A of orer [ ] t havig o zero rows, where r t < = mir,w j= w r j w w { + l[ + r w j r w r j mir,w j= w j mir,w j= w { + l[ w r j mir,w j= w j ]} w r j w j Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a, r-isjuct matrix of orer t. We the ]}. have t,, r < < r w mir,w w { + l[ w w j= r j j j= r w w { + l[ w w w r r mir,w w w r j ]}, w j ]} by Lemma.4.. Let be the smallest positive iteger such that w = is a iteger. 35

44 We have r w w w r + r by Lemma.4.5 taig s =, a w w l r w by Lemma.4.6 taig s =. Therefore, t,, r t,, r < as require. < [ + l + ] + l r { w w w + l[ w w r r < r + { + [ + l + ] + l } w ]} Recall that t,, r; z is the miimum size over all, r; z-isjuct matrices with colums. Theorem For ay positive itegers,, r a z, if = + r, the t,, r; z < z r + { + [ + l + ] + l }. Proof. For w, let A be the biary matrix of orer [ ] with r w [] [] rows a colums iexe by {D, R D, R with D R empty} a r V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, R a the colum iexe by the vector v V is if the etries of v over D are all zero a at least oe etry of v over R is oe; a 0 otherwise. 36

45 Observe that each row of A has weight mir,w j= r j + r mir,w w w w A has weight r j j j= there exists a submatrix M of A of orer [ r where z mir,w w w t < mir,w { + l[ w w ]} r + r r j j j= j w j j= z mir,w r w = mir,w w { + l[ w w w r j j= r j j j=, a each colum of w j. By the extee Stei-Lovász theorem, ] t with each row weight at least z, w j Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a, r; z-isjuct matrix of orer t. We ]}. the have t,, r; z < < z r w mir,w w { + l[ w w j= r j j j= z r w w { + l[ w w w r r mir,w w w r j ]}, w j ]} by Lemma.4.. Let be the smallest positive iteger such that w = We have is a iteger. r w w w r + r 37

46 by Lemma.4.5 taig s =, a w w l r w by Lemma.4.6 taig s =. Therefore, < [ + l + ] + l t,, r; z t,, r; z z r < { w w w + l[ w w r r < z r + { + [ + l + ] + l } as require. w ]} Recall that t,, r, s] is the miimum size over all, s out of r]-isjuct matrices with colums. Theorem For ay positive itegers,, r a s, with s r, if = + r, the t,, r, s] < s s s s { + [ + l s + ] + l }. r s Proof. For s w, let A be the biary matrix of orer [ ] with r w [] [] rows a colums iexe by {D, R D, R with D R empty} a r V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, R a the colum iexe by the vector v V is if the etries of v over D are all zero a at least s etries of v over R are oe; a 0 otherwise. mir,w r + r Observe that each row of A has weight, a each colum of A j w j j=s 38

47 mir,w w w w has weight r j j j=s submatrix M of A of orer [ r t < = mir,w j=s w r j w w { + l[ + r w j r w r j mir,w j=s w j. By the Stei-Lovász theorem, there exists a ] t havig o zero rows, where mir,w j=s w { + l[ w r j mir,w j=s w j ]} w r j w j Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a, s out of r]-isjuct matrix of orer t. We the have r w t,, r, s] < mir,w w { + l[ w w j=s r j j j=s r w w < { + l[ w w w r s r s s mir,w w s w by Lemma.4.. Let be the smallest positive iteger such that w = s We have r w s s s s w w r r s s s r j ]}, ]}. w j ]} is a iteger. by Lemma.4.5, a w w l r s w s < [ + l + ] + l s 39

48 by Lemma.4.6. Therefore, t,, r, s] t,, r, s] r < { w w w + l[ w w r s r s s < s s s s { + [ + l s + ] + l } as require. r s w s ]} Recall that t,, r, s; z] is the miimum size over all, s out of r; z]-isjuct matrices with colums. Theorem For ay positive itegers,, r, s a z, with s r, if = + r, the t,, r, s; z] < z s s s s { + [ + l s + ] + l }. r s Proof. For s w, let A be the biary matrix of orer [ ] with r w [] [] rows a colums iexe by {D, R D, R with D R empty} a r V = {v v {0, }, wtv = w} respectively. The etry of A at the row iexe by the pair D, R a the colum iexe by the vector v V is if the etries of v over D are all zero a at least s etries of v over R are oe; a 0 otherwise. mir,w r + r Observe that each row of A has weight, a each colum of j w j j=s mir,w w w w A has weight. By the extee Stei-Lovász theorem, r j j j=s there exists a submatrix M of A of orer [ ] t with each row weight at least z, r 40

49 where t < = mir,w j=s w r j z w w { + l[ + r w j z r w r j mir,w j=s w j mir,w j=s w { + l[ w r j mir,w j=s w j ]} w r j w j ]}. Note that the equality is obtaie by coutig the weight of A i two ways. It is straightforwar to show that the colums of M form a, s out of r; z]-isjuct matrix of orer t. We the have t,, r, s; z] < < z r w mir,w w { + l[ w w j=s r j j j=s z r w w { + l[ w w w r s r s s mir,w w s w r j ]}, w j ]} by Lemma.4.. Let be the smallest positive iteger such that w = s We have r w s s s s w w r r s s s is a iteger. by Lemma.4.5, a w w l r s w s < [ + l + ] + l s 4

50 by Lemma.4.6. Therefore, t,, r, s; z] t,, r, s; z] z r < { w w w + l[ w w r s r s s < z s s s s { + [ + l s + ] + l } as require. r s w s ]} 4. Bous for several selectors The otio of, m, -selectors was first itrouce by De Bois, Gasieiec a Vaccaro i [], a it was the geeralize to the otio of, m, c, -selectors []. It is iterestig to remar that the otios of, m, -selecters a, m, c, -selectors are equivalet to, m,, ; -selectors a, m, c, ; -selectors respectively. Note that upper bous for the sizes of, m, -selectors were also give i [4] by the Lovász Local Lemma. Followig similar argumets i [] a [] with a mior moificatio, upper bous for the sizes of several selectors are give below. Recall that t s, m, is the miimum size over all, m, -selectors. Theorem 4... t s, m, < m + + { + [ + l + ] + l }. m Proof. For w +, let X = {x {0, } wtx = w} a U = {u {0, } wtu = }. Moreover, for ay A U of size r, r =,...,, a ay set S [], efie E A,S = {x X : x S A}. Let M be the biary matrix of orer [ m+ ] w with rows a colums iexe 4

51 by Γ = {E A,S X A U with U =, A = m +, S [] } a X = {x {0, } wtx = w} respectively. The etry of M at the row iexe by the set E A,S a the colum iexe by the vector x X is if x E A,S ; a 0 otherwise. Observe that each row of M has weight m+ w w orer [ m+ w, a each colum of M has weight. By the Stei-Lovász theorem, there exists a submatrix M of M of m+ ] t havig o zero rows, where t < w w w m + { + l[ ]} m w = m+ w w w w { + l[ ]}. m m Note that the equality is obtaie by coutig the weight of M i two ways. It suffices to show that the matrix M of orer t forme by the colums of M is a, m, -selector, that is, ay submatrix of arbitrary colums of M cotais a submatrix with each row weight exactly oe, with at least m istict rows. Let x, x,..., x t be the t rows of M a let T = {x, x,..., x t }. Suppose cotraictorily that there exists a set S [] such that the submatrix M S of M cotais a submatrix with each row weight exactly oe, with at most m istict rows. Let u j, u j,..., u jq be such rows, with q m ; let A be ay subset of U\{u j, u j,..., u jq } of cariality A = m +, the we have T E A,S =, cotraictig the fact that M orer [ m+ ] t havig o zero rows. Hece we have t s, m, < m+ w w m w { + l[ w m ]}. is a matrix of Let be the smallest positive iteger such that w = m+ w w m =! m! m+!! m! m! w = w m m + is a iteger. We have w w

52 by Lemma.4.5 taig s = r =, a w l[ w m by Lemma.4.6 taig s = r =. Therefore, we have ] < [ + l + ] + l m as require. t s, m, t s, m, < < m+ w w m m + + w { + l[ w { + [ + l m ]} + ] + l } m Recall that t s, m, ; z is the miimum size over all, m, ; z-selectors. Theorem 4... t s, m, ; z < m + z + + m + { + [ + l + ] + l }. m Proof. For w +, let X = {x {0, } wtx = w} a U = {u {0, } wtu = }. Moreover, for ay A U of size r, r =,...,, a ay set S [], efie E A,S = {x X : x S A}. Let M be the biary matrix of orer [ m+ ] w with rows a colums iexe by Γ = {E A,S X A U with U =, A = m +, S [] } a X = {x {0, } wtx = w} respectively. The etry of M at the row iexe by the set E A,S a the colum iexe by the vector x X is if x E A,S ; a 0 otherwise. Observe that each row of M has weight m+ m+ w w M of orer [ w, a each colum of M has weight. By the extee Stei-Lovász theorem, there exists a submatrix M of m+ ] t with each row weight at least m + z +, where 44

53 t < [ m + z + ] w w w m + { + l[ ]} m w = [ m + z + ] m+ w w w w { + l[ ]}. m m Note that the equality is obtaie by coutig the weight of M i two ways. It suffices to show that the matrix M of orer t forme by the colums of M is a, m, ; z-selector, that is, ay submatrix of arbitrary colums of M cotais z isjiot submatrices with each row weight exactly oe, with at least m istict rows each. Let x, x,..., x t be the t rows of M a let T = {x, x,..., x t }. Suppose cotraictorily that there exists a set S [] such that the submatrix M S of M cotais at most z isjoit submatrices with each row weight exactly oe, with at least m istict rows. Moreover, M S cotais aother isjoit submatrix with at most m istict rows with weight exactly oe. Let u j, u j,..., u jq be such rows, with q m ; let A be ay subset of U\{u j, u j,..., u jq } of cariality A = m +, the we have T E A,S < m + z +, cotraictig the fact that M is a matrix of orer [ m+ ] t with each row weight at least m + z +. Hece we have t s, m, ; z < [ m + z + ] m+ w w w { + l[ m Let be the smallest positive iteger such that w = m+ w w m =! m! m+!! m! m! by Lemma.4.5 taig s = r =, a w l[ w w = w m + + m w m + m ]}. is a iteger. We have w w ] < [ + l + ] + l m 45

54 by Lemma.4.6 taig s = r =. Therefore, we have t s, m, ; z t s, m, ; z < [ m + z + ] m+ w w w { + l[ m as require. < w m + z + + m + { + [ + l ]} m + ] + l m } Moreover, we will also give better, but ocostructive upper bous for the sizes of, m, ; z-selectors as follows. Lemma 4... A, m, ; z-selector M is m, m + ; z-isjuct. Proof. Suppose ot. The there exist colums C, C,..., C of M such that i=m m C i \ C i z, i= that is, there exist at most z rows with row weight such that each of them hits exactly oe of the colums C m, C m+,..., C. Hece there exist at most z isjoit m submatrices of the ietity matrix I, it cotraicts the fact that M is a, m, ; z-selector. By Lemma 4.., upper bous for the sizes of, m, ; z-selectors ca be obtaie from that of m, m + ; z-isjuct matrices. Hece ocostructive upper bous for the sizes of, m, ; z-selectors are give below. Corollary 4... t s, m, ; z < z m + + { + [ + l + ] + l }. m 46

55 Proof. Put, r = m, m i t,, r; z as show i Theorem 4..6, we the have t s, m, ; z < z m + + { + [ + l + ] + l }. m Remar 4... For the case z =, i.e., t s, m, ; < m + + { + [ + l + ] + l } m was also give i Theorem 4.. followe by a costructive proof i term of the Stei-Lovász theorem. Recall that t s, m, c, is the miimum size over all, m, c, -selectors. Theorem For a = t s, m, c, <, c a m + c c + c c { + [ + l + ] + l a }. a m Proof. For c w + c, let X = {x {0, } wtx = w} a U = {u {0, } wtu = c}. Moreover, for ay A U of size r, r =,..., c, a ay set S [], efie E A,S = {x X : x S A}. Let a = c a M be the biary matrix of orer [ a a m+ ] w with rows a colums iexe by Γ = {E A,S X A U with U = a, A = a m +, S [] } a X = {x {0, } wtx = w} respectively. The etry of M at the row iexe by the set E A,S a the colum iexe by the vector x X is if x E A,S ; a 0 otherwise. Observe that each row of M has weight a m+ w c w c a a m+ w c, a each colum of M has weight. By the Stei-Lovász theorem, there exists a submatrix M of M of 47

56 orer [ a a m+ ] t havig o zero rows, where t < w w w a a m + { + l[ ]} c c a m w c a = a m+ w w a w w a { + l[ ]}. c c c a m c a m Note that the equality is obtaie by coutig the weight of M i two ways. It suffices to show that the matrix M of orer t forme by the colums of M is a, m, c, -selector, that is, ay submatrix of arbitrary colums of M cotais a submatrix with each row weight exactly c, with at least m istict rows. Let x, x,..., x t be the t rows of M a let T = {x, x,..., x t }. Suppose cotraictorily that there exists a set S [] such that the submatrix M S of M cotais a submatrix with each row weight exactly c, with at most m istict rows. Let u j, u j,..., u jq be such rows, with q m ; let A be ay subset of U\{u j, u j,..., u jq } of cariality A = a m+, the we have T E A,S =, cotraictig the fact that M is a matrix of orer [ a a m+ ] t havig o zero rows. Hece we have t s, m, c, < a a m+ w c w c a a m w { + l[ c w c Let be the smallest positive iteger such that w = c a a m+ w c w c a a m = = a! m!a m+! a! m!a m! a m + by Lemma.4.5 taig s = r = c, a w l[ c w c a a m w = c w c w c w c c a m + a a m ]}. is a iteger. We have a w c w c a m + c c + c c ] < [ + l a + ] + l a m 48

57 by Lemma.4.6 taig s = r = c. Therefore, we have t s, m, c, t s, m, c, as require. < < a a m+ w { + l[ c w c w a c a m a m + c c + w c a a m c c { + [ + l ]} + ] + l a } a m Recall that t s, m, c, ; z is the miimum size over all, m, c, ; z-selectors. Theorem For a = t s, m, c, ; z <, c a m + z + a m + c c + c c {+[+l a +]+l }. a m Proof. For c w + c, let X = {x {0, } wtx = w} a U = {u {0, } wtu = c}. Moreover, for ay A U of size r, r =,..., c, a ay set S [], efie E A,S = {x X : x S A}. Let a = c a M be the biary matrix of orer [ a a m+ ] w with rows a colums iexe by Γ = {E A,S X A U with U = a, A = a m +, S [] } a X = {x {0, } wtx = w} respectively. The etry of M at the row iexe by the set E A,S a the colum iexe by the vector x X is if x E A,S ; a 0 otherwise. Observe that each row of M has weight a m+ w w c c M of orer [ a a m+ a w c, a each colum of M has weight. By the extee Stei-Lovász theorem, there exists a submatrix M of a m+ ] t with each row weight at least a m + z +, where t < [a m + z + ] w w w a a m + { + l[ ]} c c a m w c = [a m + z + ] a a m+ w w a w w a { + l[ ]}. c c c a m c a m 49