Model Theoretic Methods in Finite Combinatorics.

Intro -- Contents -- Preface -- Application of Logic to Combinatorial Sequences and Their Recurrence Relations -- Part 1. Introduction and Synopsis -- 1. Sequences of integers and their combinatorial interpretations -- 2. Linear recurrences -- 3. Logical formalisms -- 4. Finiteness conditions -- 5. Logical interpretations of integer sequences -- Part 2. Guiding Examples -- 6. The classical recurrence relations -- 7. Functions, permutations and partitions -- 8. Trees and forests -- 9. Graph properties -- 10. Latin squares -- Part 3. C-Finite and Holonomic Sequences -- 11. C-Finite sequences -- 12. Holonomic sequences -- Part 4. Modular Recurrence Relations -- 13. DU-index and Specker index -- 14. The rôle of logic -- 15. Structures of bounded degree -- 16. Structures of unbounded degree -- References -- Spectra and Systems of Equations -- Compton's Method for Proving Logical Limit Laws -- Logical Complexity of Graphs: A Survey -- 1. Introduction -- 1.1. Basic notions and examples -- 1.2. Variations of logic -- 1.3. Outline of the survey -- 1.4. Other structures -- 2. Preliminaries -- 2.1. Notation: Arithmetic and graphs -- 2.2. A length-depth relation -- 2.3. Distinguishability vs. definability -- 3. Ehrenfeucht games -- 4. The Weisfeiler-Lehman algorithm -- 5. Worst case bounds -- 5.1. Classes of graphs -- 5.2. General case -- 6. Average case bounds -- Methods for Algorithmic Meta Theorems -- On Counting Generalized Colorings -- 1. Introduction -- 2. Prelude: two typical graph polynomials -- 3. Counting generalized colorings -- 4. SOL-polynomials and subset expansion -- 5. Standard vs FF vs Newton SOL-polynomials -- 6. Equivalence of counting φ-colorings and SOL-polynomials -- 7. MSOL-polynomials -- 8. Enter categoricity -- 9. Conclusions -- References -- Counting Homomorphisms and Partition Functions.

Some Examples of Universal and Generic Partial Orders -- Two Problems on Homogeneous Structures, Revisited -- On Symmetric Indivisibility of Countable Structures -- Partitions and Permutation Groups -- (Un)countable and (Non)effective Versions of Ramsey's Theorem -- Reducts of Ramsey Structures -- 1. Introduction -- 2. Reducts -- 3. Ramsey Classes -- 4. Topological Dynamics -- 5. Minimal Functions -- 6. Decidability of Definability -- 7. Interpretability -- 8. Complexity of Constraint Satisfaction -- 9. Concluding Remarks and Further Directions -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.