On Sets and Graphs Perspectives on Logic and Combinatorics

This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels.

Introduction

Part I: Basics

Membership and Edge Relations

Sets, Graphs, and Set Universes

Part II: Graphs as Sets

The Undirected Structure Underlying Sets

Graphs as Transitive Sets

Part III: Sets as Graphs

Counting and Encoding Sets

Random Generation of Sets

Infinite Sets and Finite Combinatorics

Appendix: Excerpts from a Referee-Checked Proof-Script