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Definable Combinatorics of Graphs and Equivalence Relations Definable Combinatorics of Graphs and Equivalence Relations Thesis by Connor Meehan In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 Defended May 22, 2018 © 2018 Connor Meehan ORCID: 0000-0002-7596-2437 All rights reserved ii ACKNOWLEDGEMENTS iii I would like to firstly thank my advisor, Alexander Kechris, for his patience, encouragement, and mentorship throughout my time as a graduate student. I would like to thank the department of mathematics at the California Institute of Technology for its surprisingly effective efforts at fostering a sense of community. I would like to thank Ronnie Chen for his constant companionship, our countless discussions (mathematical or otherwise), and his astute perspectives shared within. I owe great thanks to William Chan for profoundly increasing the enjoyment of my graduate studies, for supplying an outlet for my frustrations, and most of all, for his friendship. I would also like to thank Duncan Palamourdas for our brief and inspiring correspondence. I would like to thank my parents, Deborah and Stephen Meehan, for their support for me while I pursued my dreams. I would like to thank Cassandra Chan for showing that she cared. Lastly, I would like to thank both the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation. This research is partially supported by NSERC PGS D and by NSF grant DMS-1464475. ABSTRACT iv Let D = ¹X; Dº be a Borel directed graph on a standard Borel space X and let χB ¹Dº be its Borel chromatic number. If F0;:::; Fn−1 : X ! X are Borel functions, let DF0;:::;Fn−1 be the directed graph that they generate. It is an open problem if χB DF0;:::;Fn−1 2 f1;:::; 2n + 1; @0g. Palamourdas [25] verified the foregoing for commuting functions with no fixed points. We show here that for commuting functions with the property that there is a path from each x 2 X to a fixed point of Ð some Fj, there exists an increasing filtration X = m<! Xm such that χB DF0;:::;Fn−1 Xm ≤ 2n for each m. We also prove that if n = 2 in the previous case, then χB DF0;F1 ≤ 4. It follows that ap the approximate measure chromatic number χM ¹Dº ≤ 2n + 1 when the functions commute. n n If X is a set, E is an equivalence relation on X, and n 2 !, then define »X¼E = f¹x0; :::; xn−1º 2 X : ¹ i; jº¹i , j ! :¹xi E xjººg. For n 2 !, a set X has the n-Jónsson property if and only if for every 8 n n function f : »X¼= ! X, there exists some Y ⊆ X with X and Y in bijection so that f »»Y¼=¼ , X. Ð n A set X has the Jónsson property if and only for every function f : ¹ n2!»X¼=º ! X, there exists Ð n someY ⊆ X with X andY in bijection so that f » n2!»Y¼=¼ , X. Let n 2 !, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager n n C ⊆ X, there is some Borel A ⊆ X so that E ≤B E A and »A¼E ⊆ C. The following equivalence ! relations will be considered: E0 is defined on 2 by x E0 y if and only if ¹ nº¹ k > nº¹x¹kº = y¹kºº. ! ! 9 8 ! E1 is defined on ¹ 2º by x E1 y if and only if ¹ nº¹ k > nº¹x¹kº = y¹kºº. E2 is defined on 2 by Í 1 9 8 ! ! x E2 y if and only if f n+1 : x¹nº , y¹nºg < 1. E3 is defined on ¹ 2º by x E3 y if and only if ¹ nº¹x¹nº E0 y¹nºº. Holshouser and Jackson have shown that R is Jónsson under AD. The present 8 research will show that E0 does not have the 3-Mycielski property and that E1, E2, and E3 do not ! have the 2-Mycielski property. Under ZF + AD, 2/E0 does not have the 3-Jónsson property. Let G = ¹X; Gº be a graph and define for b ≥ 1 its b-fold chromatic number χ¹bº¹Gº as the minimum size of Y such that there is a function c from X into b-sets of Y with c¹xº \ c¹yº = ; if x G y. Then ¹ º f χ b ¹Gº its fractional chromatic number is χ ¹Gº = infb b if the quotients are finite. If X is Polish and f G is a Borel graph, we can also define its fractional Borel chromatic number χB¹Gº by restricting to only Borel functions. We similarly define this for Baire measurable and µ-measurable functions for a Borel measure µ. We show that for each countable graph G, one may construct an acyclic 0 f 0 f 0 Borel graph G on a Polish space such that χBM ¹G º = χ ¹Gº and χBM ¹G º = χ¹Gº, and similarly f f for χµ and χµ. We also prove that the implication χ ¹Gº = 2 ) χ¹Gº = 2 is false in the Borel setting. PUBLISHED CONTENT AND CONTRIBUTIONS v [3] William Chan and Connor Meehan. “Definable Combinatorics of Some Borel Equiva- lence Relations” In: ArXiv e-prints (Sep. 2017). arXiv: 1709.04567 [math.LO]. Connor Meehan participated in producing the results and writing the paper. Each author contributed equally to this work. TABLE OF CONTENTS vi Acknowledgements . iii Abstract . iv Published Content and Contributions . v Table of Contents . vi Chapter I: Introduction . 1 Chapter II: Borel Chromatic Numbers of Graphs of Commuting Functions . 4 2.1 Introduction . 4 2.2 The General Case . 6 2.3 Commuting Functions . 7 2.4 Two Commuting Functions . 16 Chapter III: Definable Combinatorics of Some Borel Equivalence Relations . 20 3.1 Introduction . 20 3.2 Basic Information . 25 3.3 !2 Has the Jónsson Property . 30 3.4 !-Jónsson Function for !2 ............................... 34 3.5 The Structure of E0 .................................. 36 ! 3.6 2/E0 Has the 2-Jónsson Property . 41 ! 3.7 Partition Properties of 2/E0 in Dimension 2 . 45 3.8 E0 Does Not Have the 3-Mycielski Property . 46 3.9 Surjectivity and Continuity Aspects of E0 ....................... 48 ! 3.10 2/E0 Does Not Have the 3-Jónsson Property . 51 ! 3.11 Failure of Partition Properties of 2/E0 in Dimension Higher Than 2 . 54 3.12 P3 Is Proper . 55 bE0 3.13 E1 Does Not Have the 2-Mycielski Property . 57 3.14 The Structure of E2 .................................. 58 3.15 E2 Does Not Have the 2-Mycielski Property . 66 3.16 Surjectivity and Continuity Aspects of E2 ....................... 68 3.17 The Structure of E3 .................................. 73 3.18 E3 Does Not Have the 2-Mycielski Property . 75 3.19 Completeness of Ultrafilters on Quotients . 76 3.20 Conclusion . 78 Chapter IV: Fractional Borel Chromatic Numbers and Definable Combinatorics . 79 4.1 Introduction . 79 4.2 Elementary Facts about the Fractional Chromatic Number . 81 4.3 The Baire Measurable Case . 83 4.4 The µ-Measurable Case . 85 4.5 Combining All Quantities . 86 f 4.6 The χB¹Gº = 2 Case . 87 Bibliography . 89 vii C h a p t e r 1 1 INTRODUCTION Descriptive set theory is a branch of set theory that focuses on sets of a particular structure definable via some constructive rules, as opposed to sets whose existence is guaranteed by some consequence of the axiom of choice. A Polish space is a topological space that is separable and completely metrizable. The prototypical example of a definable set is a Borel set: one that belongs to the smallest family of sets containing all open sets and closed under complements and countable unions. To repeat the first sentence more concretely, descriptive set theory concerns the study of definable sets in Polish spaces. Descriptive set theory is a major research area of set theory and can both be approached from and have results that relate to many other fields of mathematics, such as mathematical logic, topology, and ergodic theory. In this thesis, we focus chiefly on descriptive combinatorics. In particular, we study problems related to graphs and equivalence relations from the viewpoint of descriptive set theory, asking when certain definable objects will satisfy certain definable properties. For example, every acyclic graph can be 2-coloured, but it is possible for the minimum size of the range of a Borel colouring of an acyclic graph to be 2@0 . This thesis consists of three separate papers: Borel Chromatic Numbers of Graphs of Commuting Functions (with Konstantinos Palamourdas), Definable Combinatorics of Some Borel Equivalence Relations (with William Chan), and Fractional Borel Chromatic Numbers and Definable Combina- torics. The first two papers have been submitted for publication. For a more detailed introduction to each paper, see the introduction section of each chapter. In Chapter 2, we study a Borel graph colouring problem; namely, what is the smallest size of a finite set (if one even exists) that can be used for a Borel colouring of a directed graph generated by n Borel functions on a standard Borel space? Palamourdas [25] has previously shown that for such a digraph DF0;:::;Fn−1 , its Borel chromatic number χB¹DF0;:::;Fn−1 º is in the set 1 1;:::; 2 ¹n + 1º¹n + 2º; @0 and that if the functions Fj commute and have no fixed points, then this can be improved to f1;:::; 2n + 1; @0g. In Chapter 2, we provide a recursive structure for breaking down subdividing a Borel graph of this structure.
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