An Introduction to WQO and BQO Theory Thomas Forster, Nathan Bowler and Monika Seisenberger June 29, 2021 ii Contents 1 Background 11 1.1 WQOs and Finiteness . 11 1.2 Definitions and Notation . 18 1.2.1 Lists and streams . 19 1.2.2 Multisets . 20 1.3 Other Background . 25 2 Quasiorders 31 2.1 Lifts . 31 2.2 Lifts to other structures . 35 2.2.1 Lists and streams . 36 2.2.2 Multisets . 37 2.2.3 Finite and infinite trees . 39 2.3 Finite character . 41 2.4 How do these operations affect rank? . 43 3 WQOs 45 3.1 Ranks of WQOs . 52 3.2 The Minimal Bad Sequence Construction . 54 3.3 Kruskal's theorem . 57 3.3.1 The tree of finite bad sequences . 60 3.4 Topics for discussion . 66 3.5 Some more exercises . 66 4 Good quasiorders of finite exponent 67 4.0.1 Reconstituting . 71 4.0.2 Some exercises . 72 4.1 Rado's quasiorder . 73 4.2 Finite exponent stuff to be ironed out . 79 iii iv CONTENTS 4.3 Diestel's lemma or something like that . 85 4.4 A Refinement, october 9th 2020 . 87 5 BQOs 89 5.1 Blocks and Games . 92 5.1.1 Generating a family of blocks recursively . 95 5.1.2 Combinatorial Definition of BQO . 100 5.2 Generalising the RADO structure to higher exponents . 104 5.3 Laver's proof of Fra¨ıss´e'sconjecture . 108 5.3.1 Laver's theorem is best possible . 110 5.4 Barwise approximants to the 1-1 embedding . 111 5.4.1 The intersection of the approximants . 113 5.5 The cofinite quasiorders on the power set . 114 5.6 Well Relations . 116 5.7 Exercises on BQOs . 117 6 The Topological Approach 119 6.0.1 A letter from Adrian Mathias . 126 7 BQOs and fast-growing functions etc 129 7.0.1 Γ0 ............................. 129 7.1 Friedman's Finite Form . 131 7.1.1 Why Kruskal's theorem is so strong . 133 7.1.2 Extending and refining FFF . 134 7.2 Generalising lists to n-lists . 136 7.2.1 n-templates . 136 7.2.2 n-template Morphisms . 137 7.2.3 n-lists . 137 7.2.4 Recursive definition of a datatype of n-lists? . 139 7.2.5 n-stretching (of n-lists) . 139 7.2.6 Inductive definition of n-stretching . 139 7.3 James's proof that you can't kill BQOs by forcing . 140 7.4 Things to get straight one day . 144 7.4.1 The least fixed point for + is the rank-comparison relation . 153 8 Answers to Selected Exercises 171 CONTENTS 1 9 Useful leftovers to be incorporated in due course 181 9.1 Hall's marriage problem|the Infinite case Pinched from Damerell-Milner 1974 . 186 9.2 Miscellaneous topics . 199 9.2.1 The 1-1-embedding . 203 9.2.2 Gap-embeddings . 212 10 Miscellaneous stuff 215 2 CONTENTS Preface This book has grown out of lectures for a graduate (\Part III") course enti- tled `Logic and Combinatorics' given at the University of Cambridge. Since motivations for interest in WQO theory are various I should perhaps explain that that course arose from my desire to show to my students a beautiful result of J.B. Kruskal and Harvey Friedman's which goes some way to ex- plaining why there should be courses with this title in the first place. The result in question is \FFF": Friedman's Finite Form|of Kruskal's theorem on the wellquasiorderings of trees. Logicians have known ever since the days of G¨odel'sIncompleteness theorems that for any axiomatic system of arithmetic there are logically simple assertions of arithmetic not provable in that system, but until the advent of FFF no examples were known that were mathematically natural. FFF arguably still remains the most natu- ral and pleasing example of such a formula. (The closest competitor, the Paris-Harrington formula, was also in that course, but did not make it into this book because it doesn't involve WQOs or BQOs). I still remember the talk where I first heard it, given by my Doktorvater, Adrian Mathias, in the early 1980's. Others will have different reasons for interest. Theoretical computer sci- entists are interested in WQO and BQO theory because it underpins their craft of proving termination of algorithms. (Indeed my sole original con- tribution to BQO theory appeared in the Journal of Theoretical Computer Science.) Finally, people interested in descriptive set theory will have had their attention drawn to BQO theory by the important and influential work of Steve Simpson ([74]), which gave a very smooth treatment of BQOs that made clear the connections with infinite Ramsey theory. This book came to be written in a way that I suspect many textbooks are written. It is the book that I wish I had had to hand when I embarked on my attempt to understand BQO theory. I do not claim to be an expert on BQO theory, and this book is not the definitive pronouncement of a master, but something which with luck may be very nearly as useful, namely the 3 4 CONTENTS log-book of the labours of a journeyman|with the false starts and fruitless errors edited out. Being by nature a lazy reader, I have worked most of this out for myself, with the help of clues I could find in the literature. I have rediscovered a lot of results that i could have found in the literature had i read it properly. Serves me right. I have cited everything I have read, and a great deal that I haven't. I cannot even guarantee that the proofs I supply of cited results are the cited proofs|being, as i have said, a lazy reader. Some of them may even be original. Most of the proofs I supply are proofs I found myself, and although I have given credit to other authors where I know it to be due, I make no claims of priority for unattributed results. I owe a great debt to colleagues with whom i thrashed out some of the details: Harold Simmons (now sadly no longer with us) and my students [. ]. It is a pleasure to be able to thank my long-suffering expert correspondents Steve Simpson, Richard Laver (also, sadly, no longer with us) Monika Seisenberger (who has succumbed to my blandishments and agreed to be a coauthor) and Alberto Marcone, who patiently and courteously answered the questions|many of them no doubt quite daft|with which I plied them during my attempts to teach myself this material. One of my reasons for writing this book is to ensure, by setting down in writing (some of) what I have learned from them, that they will be in future slightly safer than they evidently were in the past from importunacies like mine. Finally I owe a special debt of gratitude to my Doktorvater Adrian Math- ias. In about 1983 he gave a lecture at a LOGFIT meeting in Leeds in which he presented Nash-Williams' proof of Kruskal's theorem and gave a proof of Friedman's finite form of Kruskal's theorem, then hot off the press. I recog- nised it immediately as something I wanted to know more about. Adrian's talk plunged me head-first into a programme of self-improvement of which this book is the result. I would like to be able to say that I wouldn't have done it but for him, but this material is so fascinating that I suspect that it would have grabbed me|somehow-or-other|in any event. But Adrian was the proximate cause. Must resolve to use `Q' rather than `X' for the generic variable for a quasiorder. Stuff to fit in Blocks are sets of finite sequences of naturals. Thus one can sensibly ask whether or not a block is decidable/semidecidable. No-one seems to have CONTENTS 5 done so. I've been thinking about unfoldings again. I am currently very struck by the thought that the derivative of a quasiorder R is a bit like the square of R \with certificates”. But that raises the point that one can obtain a three-place relation from R and S. Let us write R⊕S for fhx; y; zi : hx; yi 2 S ^ hy; zi 2 Rg Does every BQO that is a set of triples arise as R ⊕ S for two quasiorders R and S. And another thing! As Imre and Gareth have crsiply shown, <IN is not the square of any relation, and their proof shows that no wellordering is a square. What about the strict part of a WQO? Rod Downey has just said something to me that has focussed all my thoughts about Nash-Williams' proof of Kruskal wot Adrian showed me all those years ago. Rod had a proof which is in some sense the correct presentation of the specific instance of the general result that is NW's proof of Kruskal. I now understand the generalisation. Rod's proof is the following. Dust off for future use the observation that every homomorphic image of a good sequence is a good sequence, so a preimage of a bad sequence is also a bad sequence. Let wombat be a recursive datatype of finite character, so that a wombat is built up from the null wombat by finitely many invocations of constructors| of which there are only finitely many. NW presents trees and lists in this way.