Finite and Infinite Combinatorics in Sets and Logic

Transcription

1 Finite and Infinite Combinatorics in Sets and Logic

2 NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scient ific Affairs Division A Life Sciences B Physics C Mathematical and Physical Sciences D Behavioural and Social Sciences E Applied Sciences F Computer and Systems Sciences G Ecological Sciences H Cell Biology I Global Environmental Change Plenum Publishing Corporation London and New York Kluwer Academic Publishers Dordrecht, Boston and London Springer-Verlag Berlin, Heidelberg, New York, London, Paris and Tokyo NATO PCO-DATA BASE The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than contributions from internat ional scientists published in aii sections of the NATO ASI Series. Access to the NATO-PCO-DATA BASE is possible in two ways: - via online FILE 128 (NATO-PCO-DATA BASE) hosted by ESRIN, Via Galileo Galilei, ' Frascati, Italy. - via CD-ROM "NATO-PCO-DATA BASE" with user-friendly retrieval software in English, French and German ( WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Series C: Mathematical and Physical Sciences - VoI. 411

3 Finite and Infinite Combinatorics in Sets and Logic edited by N.W. Sauer, R.E. Woodrow and B. Sands Department of Mathematics and Slalislics, The University of Calgary, Calgary, Alberta, Canada Springer-Science+Business Media, B.V.

4 Proceedings of the NATO Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic Banff, Alberta, Canada April 21 - May 4, 1991 A C.I.P. Catalogue record for this bookis available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper AII Rights Reserved 1993 Springer Science+Bus iness Media Oordrecht Originally published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover 1st edition 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 CONTENTS Preface... ix Participants xi Scientific Programme... xiii J.E. Baumgartner, A. Hajnal, S. Todorcevic. Extensions of the Erdos-Rado Theorem... 1 A. Bialostocki. Zero Sum Trees: A Survey of Results and Open Problems R. Bonnet. On Superatomic Boolean Algebras A.B. Carson. Representing Rings Using Continuous Functions G.L. Cherlin. Homogeneous Directed Graphs P. Erdos, A. Hajnal, J.A. Larson. Ordinal Partition Behavior of Finite Powers of Cardinals D.M. Evans. Some Subdirect Products of Finite Nilpotent Groups G. Hahn, J. Siran. Three Remarks on End-Faithfulness A. Hajnal. True Embedding Partition Relations R. Halin. Lattices Related to Separation in Graphs H. Harborth. Ramsey Numbers for Sets of Five Vertex Graphs With Fixed Number of Edges E. Hrushovski. Finite Structures With Few Types P. Ille. Recognition Problem in Reconstruction for Decomposable Relations W. Imrich, N. Sauer, W. Woess. The Average Size of Nonsingular Sets in a Graph J.A. Larson. Some Canonical Partition Ordinals D. Lascar. The Group of Automorphisms of a Relational Saturated Structure H. Lefmann, V. Rodl. On Canonical Ramsey Numbers for Coloring Three-Element Sets v

6 vi D. Macpherson. Large Subgroups of Infinite Symmetric Groups A.H. Mekler. Homogeneous Partially Ordered Sets E.C. Milner. Cardinal Representations S. Neufeld, R.J. Nowakowski. A Vertex-to- Vertex Pursuit Game Played With Disjoint Sets of Edges M. Pouzet. Graphs and Posets With no Infinite Independent Set I. Rival. Problems About Planar Orders T.J. Schneider. Supers table and Unstable Theories of Order.' S. Shelah. Advances in Cardinal Arithmetic S. Todorcevic. Conjectures of Rado and Chang and Cardinal Arithmetic P. Winkler. Random Structures and Zero-One Laws N. Zaguia. Isotone Maps: Enumeration and Structure Problem Sessions

7 BACK ROW (LEFT TO RIGHT) P. Ille, G. Cherlin, B. Rothschild, W.S. Shi, R. Diestel, S. Starchenko, C. Laflamme, M. Rabus, R. Halin, N. Sauer, B. Hart. THIRD ROW (LEFT TO RIGHT) W. Brierley, A. Mekler, P. Winkler, C. Lam, B. Li, H. Wang, J. Truss, D. Lascar, R. Nowakowski, D. Evans, J-M. Brochet, M. Giraudet, D. Delhomme. SECOND ROW (LEFT TO RIGHT) 1. Rival, R. Schipperus, J. Schonheim, B. Sands, A. Carson, H. Harborth, E.C. Milner, P. Erdos, A. Hajnal, D. Haskell, S. Gunay, M. Pouzet, T. Goddard. FIRST ROW (LEFT TO RIGHT) D. Duffus, H. Kierstead, N. Zaguia, R. Bonnet, S. Shelah, D. Gunderson, J. Baumgartner, J. Siran, C. Toffalori. vii

8 PREFACE This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, set theory, graph theory and logic. It used to be that infinite set theory, finite combinatorics and logic could be viewed as quite separate and independent subjects. But more and more those disciplines grow together and become interdependent of each other with ever more problems and results appearing which concern all of those disciplines. I appreciate the financial support which was provided by the N.A.T.O. Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada and the Department of Mathematics and Statistics of the University of Calgary. 11l'te meeting on Finite and Infinite Combinatorics in Sets and Logic followed two other meetings on discrete mathematics held in Banff, the Symposium on Ordered Sets in 1981 and the Symposium on Graphs and Order in The growing inter-relation between the different areas in discrete mathematics is maybe best illustrated by the fact that many of the participants who were present at the previous meetings also attended this meeting on Finite and Infinite Combinatorics in Sets and Logic. The main topics of this meeting illustrating the cohesiveness of finite and infinite discrete mathematics dealt with finite and infinite graphs, partitions of infinite graphs and rays, random graphs, random structures, homogeneous structures and graphs, the partition calculus and Ramsey theory, partition regular systems, coloring and partition problems, finite and infinite partial orders, the use of general logical constructions in discrete mathematics and stability theory, relational structures and Boolean algebras, pseudo finite structures, cardinal arithmetic and representation. The papers in this volume give a good account of the topics mentioned in the previous paragraph. They mirror the talks presented during the conference and highlight some new research, but are also intended to allow non-specialists an entrance into the area under discussion and are generally of a more survey type nature. For students interested in topics somewhere in between the theory of relations, logic, infinite set theory and finite combinatorics the present proceedings provide good first reading on those topics and demonstrate, ix

9 x due to the interconnectedness of the articles, that here is a cohesive subject very worthwhile of further study. There were lively problem and discussion sessions in the evening which led to several new results published in this volume. I am very glad and proud that we could include the results of successful cooperation initiated by the opportunity for discourse presented at the meeting. Of course this necessitated on the other hand that we had to be forgiving with deadlines and may partially explain the late publication of this volume. E.C. Milner has always been interested in both finite and infinite combinatorial mathematics. In this sense the topic of this conference and the participants have been old friends of his in whose honor we organized this meeting. I know of many mathematicians who are grateful for the unselfish help he has provided in improving their results both scientifically and stylistically. I certainly count myself as one of them. I am particularly grateful to R Woodrow and B. Sands, the co-editors of this volume, for their help without which I would have found it almost impossible to organize the meeting and prepare the proceedings for publication. I also received help from the Scientific Affairs Division of N.A.T.O. and support in the selection and notification of the main participants from M. Pouzet and A. Lachlan. Several of the participants lent a helping hand, just to mention a few of them, RK. Guy, D. Duffus and E. Milner. The help of G. Vezina, R Maltby and J. Longworth in typing as well as the very enthusiastic help of D. Gunderson is greatly appreciated. Special thanks go to Bianca Sauer for enduring my preoccupation for long periods of time. Calgary, Canada, March 5, 1993 Norbert Sauer

10 LIST OF PARTICIPANTS Brian Alspach (Canada) Joseph Barback (U.S.A.) James E. Baumgartner (U.S.A.) Arie Bialostocki (U.S.A.) Robert Bonnet (France) William Brierley (Canada) Jean-Michel Brochet (France) Andrew B. Carson (Canada) Gregory L. Cherlin (U.S.A.) Jacinta Covington (Australia) Walter Deuber (Germany) Christian Delhomme (France) Reinhard Diestel (Germany) Dwight Duffus (U.S.A.) Paul Erdos (Hungary) David M. Evans (U.K.) Michele Giraudet (France) Ted Goddard (Canada) Suleyman Gunay (Turkey) David Gunderson (U.S.A.) Richard K. Guy (England) Gena Hahn (Canada) Andras Hajnal (Hungary) Rudolf HaHn (Germany) Heiko Harborth (Germany) Bradd Hart (Canada) Egbert Harzheim (Germany) Deirdre Haskell (U.K.) Katherine Heinrich (Canada) Peter Horak (Canada) Ehud Hrushowski (U.S.A.) Pierre Ille (France) Heinz Jung (Germany) Hal A. Kierstead (U.S.A.) Claude Laflamme (Canada) Clement Lam (Canada) Jean A. Larson (U.S.A.) Daniel Lascar (France) Hanno Lefmann (Germany) Boyu Li (Canada) Dugald Macpherson (U.K.) Adrian R.D. Mathias (U.S.A.) Alan Mekler (Canada) Cheryl Miller (U.S.A.) Eric C. Milner (Canada) William Mitchell (U.S.A.) xi

11 xii Rognvaldur G. Moller (U.K.) Charles Morgan (U.S.A.) Richard J. Nowakowski (Canada) Norbert Polat (France) Maurice Pouzet (France) Mariusz Rabus Ivan Rival (Canada) Bruce Rothschild (U.S.A.) Bill Sands (Canada) Norbert Sauer (Canada) Rene Schipperus (Canada) Todd J. Schneider (U.S.A.) Jochanan Schonheim (Israel) Saharon Shelah (Israel) Jozef Siran (Czechoslovakia) Sergei Starchenko (U.S.S.R.) Juris Steprans (Canada) Michael G. Stone (U.S.A.) Stevo Todorcevic (Canada) Carlo Toffalori (Italy) J.K. Truss (U.K.) Boban Velickovic (U.S.A.) Edward T.R. Wang (Canada) Rong Wang (Canada) Shang-zhi Wang (Canada) Peter M. Winkler (U.S.A.) K.C. Wong (Canada) Carol Wood (U.S.A.) Robert E. Woodrow (Canada) Nejib Zaguia (Canada)

12 PROGRAMME FOR MONDAY APRIL 22. 9:25 a.m. N. Sauer, Opening comments 9:30-10:30 a.m. R. Halin-1Tee Partitions of Infinite Graphs 11:00-12:00 p.m. P. Winkler-Random Structures 2:00-4:00 p.m. Special Session: Relational Structures (Chairman: M. Pouzet) 2:00-2:30 p.m. P. Ille-Indecomposable Relations 2:40-3:10 p.m. D. Evans-Finite Covers of No-Categorical Structures 3:20-3:50 p.m. D. Haskell-A Transfer Theorem in Constructiv~ p-adic Algebra 4:30-5:30 p.m. G. Cherlin-Classification of Homogeneous Directed Graphs PROGRAMME FOR TUESDAY APRIL 23. 9:30-10:30 a.m. R. HaHn-Lattices of Cuts in Graphs 11:00-12:00 p.m. P. Winkler-Random Structures 2:00-4:00 p.m. Special Session: Graphs (Chairman: G.W. Sands) 2:00-2:20 p.m. B.L. Rothschild-Numbers of Vertices and Edges Which Must Occur in Induced Subgraphs 2:25-2:45 p.m. J. Siran-Discrepancy Versus Size 2:50-3:20 p.m. R. Moller-Ends of Graphs 3:25-3:45 p.m. G. Hahn-A Nice Proof of Halin's Theorem 3:50-4:10 p.m. R. Nowakowski-Vertex to Vertex Pursuit on Disjoint Sets of Edges 4:30-5:30 p.m. G. Cherlin-Classification of Homogeneous Directed Graphs 8:00-9:30 p.m.-problem session in the Solarium bar: Infinite Graphs (Chairman: M. Pouzet) xiii

13 xiv PROGRAMME FOR WEDNESDAY APRIL 24. 9:30-10:30 a.m. J. Baumgartner-Aspects of the Partition Calculus 11:00-12:00 p.m. 1. Rival-Order, Ice Flow and Invariance 12:00-Conference Photo 2:00-4:00 p.m. Special Session: Partial Orders (Chairman: D. Duffus) 2:00-2:40 p.m. H.A. Kierstead-Recursive and/or On-line Coloring of Co-comparability Graphs 2:45-3:25 p.m. N. Zaguia-On Order Preserving Maps and Perpendicular Orders 3:30-4:10 p.m. S. Todorcevic-Conjectures of Rado and Chang PROGRAMME FOR THURSDAY APRIL 25. 9:30-10:30 a.m. J. Baumgartner-Aspects of the Partition Calculus 11:00-12:00 p.m. M. Pouzet-From Relational Structures to Ordered Sets 2:00-4:00 p.m. Special Session: Homogeneous Structures (Chairman: N.W. Sauer) 2:00-2:30 p.m. J. Covington-Maximal Subgroups of Infinite Symmetric Groups 2:40-3:10 p.m. J.K. Truss-Generic Automorphisms of Homogeneous Structures 3:20-3:50 p.m. C. Wood-Partially Homogeneous Posets 4:30-5:30 p.m. Lascar D.-Automorphism Groups of Relational Structures 8:00-9:30 p.m.-problem session in the Solarium bar: Finite Combinatorics (Chairman: H. Lefmann)

14 xv PROGRAMME FOR FRlDAY APRlL 26. 9:30-10:30 a.m. M. Pouzet-From Relational Structures to Ordered Sets 11:00-12:00 p.m. R. Bonnet-On Superatomic Boolean Algebras 2:00-4:00 p.m. Special Session: Algebraic Structures (Chairman: M.G. Stone) 2:00-2:25 p.m. A. Carson-The Characterization of Function Rings 2:30-2:55 p.m. E.T.H. Wang-Expressing Positive Integers as Sums of Consecutive Positive Integers 3:00-3:25 p.m. C. Laflamme-A Few Sigma Ideals of Measure Zero Sets 3:30-3:55 p.m. B. Velickovic-On the Deterministic Approximation of DNF Formulas 4:30-5:30 p.m. D. Lascar-Automorphism Groups of Relational Structures PROGRAMME FOR SATURDAY APRlL 27. 9:30-10:30 a.m. E. Hrushovski-Pseudo Finite Structures 11:00-12:00 p.m. H. Lefmann-On Canonical Ramsey Numbers for Coloring :i-element Sets 2:00-4:00 p.m. Special Session: Infinite Combinatorics (Chairman: E.C. Milner) 2:00-2:30 p.m. C. Morgan-Morasses and Augmentations 2:35-3:05 p.m. J. Steprans-Cardinal Invariants Associated With Splitting Finite Sets 3:05-3:35 p.m. J-M. Brochet-Infinite Ordered Sets Spanned by Finitely many Chains 4:30-5:30 p.m. H.D. Macpherson-Combinatorial Questions on Infinite Permutation Groups 6:00-9:30 p.m. Banquet PROGRAMME FOR SUNDAY APRlL 28. There were three hikes organized by M.G. Stone, D. Macpherson and R.K. Guy and an excursion to the icefields.

15 xvi PROGRAMME FOR MONDAY APRlL 29. 9:3(}---1O:30 a.m. E.C. Milner-Cardinal Representation 11:0(}---12:00 p.m. P. Erdos-Some of my Favourite Problems in Finite and Infinite Combinatorics 2:0(}---4:00 p.m. Special Session: Finite Structures (Chairman: R.K. Guy) 2:0(}---2:30 p.m. N. Polat-A Compactness Theorem for Perfect Matching in Matroids 2:40-3:10 p.m. J. Schonheim-Covering Groups by Cosets,,by Conjugacy Classes 3:2(}---3:50 p.m. C. Delhomme-A Projection Property for Graphs of Girth Larger Than or Equal to Five 4:3(}---5:30 p.m. H.D. Macpherson-Combinatorial Questions on Infinite Permutation Groups 8:0(}---9:30 p.m.-problem session in the Solarium bar: Homogeneous Structures and Permutation Groups (Chairman: D. Evans) PROGRAMME FOR TUESDAY APRlL 30. 9:3(}---10:30 a.m. E.C. Milner-Cardinal Representation 11:00-12:00 p.m. R. Diestel-Graph Decomposition: A Study in Infinite Graph Theory 2:0(}---4:00 p.m. Special Session: Ramsey Theory (Chairman: H. Lefmann) 2:0(}---2:30 p.m. A. Bialostocki-Zero Sum 'I'rees: A SU1"/Jey of Results and Open Problems 2:35-3:05 p.m. H. Harborth-Ramsey Numbers For Five Vertex Graph Sets with Fixed Number of Edges. 3:1(}---3:40 p.m. D. Duffus-2-Coloring Maximal Chains in Ordered Sets 4:3(}---5:30 p.m. S. Shelah-Cardinal Arithmetic

16 xvii PROGRAMME FOR WEDNESDAY MAY 1. 9:30-10:30 a.m. A. Hajnal-1rue Embedding Partition Relations 11:00-12:00 p.m. S. Shelah-Cardinal Arithmetic 2:00-4:00 p.m. Special Session: Logic and Stability (Chairman: R.E. Woodrow) 2:00-2:30 p.m. S. Starchenko-On Strongly Abelian Stable Varieties 2:40-3:10 p.m. B. Hart-Superstable Quasi-varieties 3:20-3:50 p.m. T.J. Schneider-Stable Theories of Order 8:00-9:30 p.m.-problem session in the Solarium bar. Miscellaneous (Chairman: H. Harborth) PROGRAMME FOR THURSDAY MAY 2. 9:30-10:30 a.m. A. Hajnal-1rue Embedding Partition Relations 11:00-12:00 p.m. W.A. Deuber-On Partition Regular Systems of Equations 2:00-4:00 p.m. Special Session: Infinite Permutation Groups (Chairman: H.D. Macpherson) 2:00-2:30 p.m. O.V. Belegradek-Model Theory of Unitriangular Groups Over Rings 2:40-3:10 p.m. M. Giraudet-Automorphism Groups of Cyclic Orders 3:20-3:50 p.m. A.R.D. Mathias-The Status of the Axiom of Determinacy 4:30-5:30 p.m. J. Larson-Ordinal Partition Relations and Infinite Exponents PROGRAMME FOR FRlDAY MAY 3. 9:30-10:30 a.m. A. Mekler-Homogeneous Posets 11:00-12:00 p.m. W.A. Deuber-On Partition Regular Systems of Equations 5:00-9:30 p.m. Barbecue.