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Zeeb Road, Ann Arbor, Ml 48106 SOME RESULTS RELATED TO A CONJECTURE OF CHVATAL DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Dezsti Miklos, M.S. The Ohio State University 1986 Dissertation Committee: Approved by Professor D. K. Ray-Chaudhuri Professor T. Dowling Professor G. N. Robertson Adviser Department of Mathematics ACKNOWLEDGMENTS M y research on the topic of this dissertation began in Hungary, while I was a student, and later with a colleague of Professor G. Katona and was finished (as far as research can ever be finished) at the Ohio State University under the supervision of Professor D. K. Ray-Chaudhuri. I wish to express my deepest gratitude for their valuable assistance and patience. I wish also to thank the members of my dissertation committee, Professors T. Dowling and G. N. Robertson, as well as the entire combinatorial community of the Department of Mathematics for maintaining a pleasant working environment. Among them, I would like to mention Professor E. Bannai, from whom I learnt about a new topic of combinatorics. I offer my sincere thanks to my fellow student, Ms. Gail Gill, who read my dissertation and corrected several English (and sometimes not only English) mistakes. VITA October 28, 1957 ....................... Bom - Budapest, Hungary 1982 ......................................... M. S., Etitvos University, Budapest 1982 - 1984 ............................... Junior Fellow, Hungarian Academy of Sciences, Budapest, Hungary 1984 - .........................................Junior Research Fellow, Mathematical Institute of the Hungarian Academy of Sciences (on leave of absence) 1984 - 1986 ............................... University Fellow and Teaching Assistant (one year), The Ohio State University, Columbus, Ohio PUBLICATIONS [1] D. E. Daykin, A. J. W. Hilton and D. Miklos, Pairings from down-sets and U p :SStS in distributive lattices. J. Combin. Theory (A), 34(1983), pp. 215-230. [2] D. Miklos, Great intersecting families of edges in hereditary hypergraphs. Discr. Math., 48(1984), pp.95-99. [3] D. Miklos, Linear binary codes with intersection properties. Discr. Appl. Math., 9(1984), pp.187-196. FIELDS OF STUDY Major field: Mathematics Studies in Combinatorics: Professor D. K. Ray-Chaudhuri iii TABLE OF CONTEST ACKNOWLEDGMENTS............................................................................................... ii V ITA .................................................................................................................................... iii LIST OF TABLES ...............................................................................................................vi LIST OF FIGURES .......................................................................................................... vii INTRODUCTION ................................................................................................................. 1 Chapter Page 1. PRELIMINARIES.....................................................................................................4 1.1 Conventions 4 1.2 Definitions, notations 4 1.3 Early results 7 1.4 Chvatal's theorem 10 1.5 Berge's theorem 14 1.6 Sterboul’s theorem 22 1.7 Some other results 25 2. ANOTHER PROOF AND COMBINATORIAL GENERALIZATIONS OF BERGE'S THEOREM ...................................................................................... 28 2.1 Introduction 28 2.2 Another proof of Berge's theorem 31 2.3 Some combinatorial generalizations of Berge's theorem 33 2.4 Some other generalizations for hereditary multi- hypergraphs 38 3. A GENERALIZATION OF BERGE’S THEOREM FOR DISTRIBUTIVE L A TTIC E S ................................................................................................................41 3.1 Statement of the main result 41 3.2 An algebraic lemma 45 3.3 Proof of the main result 47 3.4 Generalization of Chvatal's conjecture for distributive lattices 60 Chapter Page 4. MAXIMUM INTERSECTING FAMILIES IN A HEREDITARY HYPERGRAPH IN THE EXTREME CASE ....................................................... 63 4.1 Introduction 63 4.2 Description of the maximum intersecting families in H 64 4.3 Proofs of Theorems 4.2 and 4.3 66 4.4 The generalizations for hereditary multi-hypergraphs 73 5. A MORE GENERAL CONJECTURE................................................................ 77 6. THE INFINITE CASE ....................................................................................... 81 6.1 Introduction 81 6.2 If there is an intersecting subfamily of H of infinite cardinality 84 6.3 The general infinite case 86 LIST OF REFERENCES.................................................................................................. 90 v LIST OF TABLES Table Page 1. Pairings from D and the bijection 9 ....................................................................29 2. Pairings from D' and the bijection <p' ...................................................................30 3. The bijections f\ and f z ......................................................................................... 43 LIST OF FIGURES Figure Page 1. Pairings from D .......................................................................................... 29 2. Pairings from D '.......................................................................................... 30 3. For the proof of Lemma 2.2 32 4. For the proof of Theorem 3.1 51 5. The lattice K .............................................................................................. 62 6. For the proof of Theorem 4.2 67 7. T, M and M' in S ...................................................................................68 8. For the proof of A2cS\T ....................................................................... 69 9. The choice of A and B ......................................................................... 70 INTRODUCTION Let S be a finite set and H cP(S) be a family of subsets of S satisfying the hereditary property, i.e., if A cB e// , then A tH as well. H is then called a hereditary hypergraph. We denote the maximum cardinality of the pairwise intersecting subfamilies of H by u>(H ) and the maximum degree of the elements of S in H by d (H ). Chvatal in 1972 conjectured that for a hereditary hypergraph H on a finite set w(LT) = d (H ) holds. This conjecture is the main core of the dissertation. In Chapter 1, we give a list of the earlier results related to the conjecture. There will be given detailed proofs for Berge's, Chvatal's, and Sterboul's results ([2], [5], and [23], respectively). Among them, the following theorem of Berge, which is given in a different form here to reduce the number of the necessaiy definitions, will play an important role throughout Chapters 2, 3, and 4. Theorem 1.12 (Berge [2]) If H is a hereditary hypergraph, then the sets from H (if |H | is even) or from H \{ 0 } (if |H | is odd) can be partitioned into pairs, such that the intersection of the two sets of a pair is empty. Chapter 2 is devoted to several generalizations of Berge’s theorem. The whole chapter, together with Chapters 3 and 4, is based on a different proof of the above theorem, which is also given here. We give the generalization of Berge's theorem, as well as some other results, for hereditary multi-hypergraphs. These results encourage us to generalize Chvatal's conjecture for hereditary multi-hypergraphs. The main result of Chapter 3, Theorem 3.1, generalizes Berge's theorem for distributive lattices with polarity. An algebraic lemma, Lemma 3.5, helps us to use combinatorial rather than algebraic methods to prove this theorem. At the end of the chapter, we give the following conjecture,