Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous n-tournaments.
Cherlin, Gregory L.
Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous n-tournaments. - 1st ed. - 1 online resource (183 pages) - Memoirs of the American Mathematical Society ; v.131 . - Memoirs of the American Mathematical Society .
Intro -- Contents -- Introduction -- Chapter 1. Results and Open Problems -- 1.1. Homogeneous structures -- 1.2. A survey of work on homogeneous structures -- 1.3. Amalgamation classes -- 1.4. Languages, strong amalgamation, generincation, and Ramsey's theorem -- 1.5. Classification theorems -- 1.6. Open problems -- Chapter 2. Homogeneous 2-tournaments -- 2.1. A catalog -- 2.2. Restricted homogeneous 2-tournaments -- 2.3. Sources and sinks -- 2.4. Constrained 2-tournaments -- 2.5. Unconstrained 2-tournaments -- Chapter 3. Homogeneous n-tournaments -- 3.1. Introduction -- 3.2. Hypercritical and small 3-tournaments -- 3.3. The critical case -- 3.4. Two embedding lemmas -- 3.5. Polarized n-tournaments -- 3.6. Embedding polarized 3-tournaments -- 3.7. Some special cases -- 3.8. The general case -- Chapter 4. Homogeneous symmetric graphs -- 4.1. The theorem of Lachlan and Woodrow -- 4.2. The main ingredients -- 4.3. Structure of the proof -- 4.4. Steps 7, 5, 8. Proof of the Main Theorems -- 4.5. Step 1, Proposition 10: adding K(2) -- 4.6. Step 1, Proposition 11: the operation H[sup(+)] -- 4.7. Step 1, Propositions 12 and 13: realization of 1-types -- 4.8. Step 2. Theorem 4.8: a, b, K -- 4.9. Step 6. Theorem 4.9.n: extending direct sums -- 4.10. Step 3. Theorem 4.6 -- 4.11. Step 4. Theorem 4.7 -- Chapter 5. Homogeneous directed graphs omitting I[sub(∞)] -- 5.1. A catalog of homogeneous directed graphs -- 5.2. The graph P(3) -- 5.3. The theorem -- 5.4. The major steps in the proof -- 5.5. Proof of the Main Theorem, Part 1 -- 5.6. Proof of the Main Theorem, part 2, n > -- 2 -- 5.7. Case 2.1 of the Main Theorem -- 5.8. Propositions 14 and 15 -- Chapter 6. Propositions 16 to 20 and MT 2.2 -- 6.1. Proposition 16: simple configurations -- 6.2. Proposition 17: induction on n -- 6.3. Proposition 18: extending I[sub(n)] -- 6.4. Proposition 19: ([sup(p)]y,y'). 6.5. Proposition 20: ([sup(p)]y,y[sup(⊥)]) -- 6.6. Toward MT 2.2 -- 6.7. Lemma 6.3 -- 6.8. Lemma 6.4 -- 6.9. Lemma 6.5 -- 6.10. Lemma 6.6 -- Chapter 7. Homogeneous directed graphs embedding I[sub(∞)] -- 7.1. The classification theorem -- 7.2. The main ingredients -- 7.3. Structure of the proof -- 7.4. Steps 4, 6, 7. The Main Theorem -- 7.5. Step 1. Proposition 24: P[sub(3)] -- 7.6. Step 1, Proposition 25: adding L(2) -- 7.7. Step 1, Proposition 26: the operations ± -- 7.8. Step 1, Propositions 27 and 28: some 1-types -- Chapter 8. Theorems 7.6-7.9 -- 8.1. Step 2. Theorems 7.6 and 7.7 -- 8.2. Step 5. Theorem 7.9.T: extending a direct sum -- 8.3. Step 3. Theorem 7.8, 1-types over sums -- 8.4. Theorem 7.8, conclusion -- Appendix: Examples for richer languages -- Bibliography -- Index of Notation -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.
9781470402105
Directed graphs.
Tournaments (Graph theory).
Model theory.
Ramsey theory.
Permutation groups.
Electronic books.
QA166.15 -- .C44 1998eb
510 s;511/.5
Classification of Countable Homogeneous Directed Graphs and Countable Homogeneous n-tournaments. - 1st ed. - 1 online resource (183 pages) - Memoirs of the American Mathematical Society ; v.131 . - Memoirs of the American Mathematical Society .
Intro -- Contents -- Introduction -- Chapter 1. Results and Open Problems -- 1.1. Homogeneous structures -- 1.2. A survey of work on homogeneous structures -- 1.3. Amalgamation classes -- 1.4. Languages, strong amalgamation, generincation, and Ramsey's theorem -- 1.5. Classification theorems -- 1.6. Open problems -- Chapter 2. Homogeneous 2-tournaments -- 2.1. A catalog -- 2.2. Restricted homogeneous 2-tournaments -- 2.3. Sources and sinks -- 2.4. Constrained 2-tournaments -- 2.5. Unconstrained 2-tournaments -- Chapter 3. Homogeneous n-tournaments -- 3.1. Introduction -- 3.2. Hypercritical and small 3-tournaments -- 3.3. The critical case -- 3.4. Two embedding lemmas -- 3.5. Polarized n-tournaments -- 3.6. Embedding polarized 3-tournaments -- 3.7. Some special cases -- 3.8. The general case -- Chapter 4. Homogeneous symmetric graphs -- 4.1. The theorem of Lachlan and Woodrow -- 4.2. The main ingredients -- 4.3. Structure of the proof -- 4.4. Steps 7, 5, 8. Proof of the Main Theorems -- 4.5. Step 1, Proposition 10: adding K(2) -- 4.6. Step 1, Proposition 11: the operation H[sup(+)] -- 4.7. Step 1, Propositions 12 and 13: realization of 1-types -- 4.8. Step 2. Theorem 4.8: a, b, K -- 4.9. Step 6. Theorem 4.9.n: extending direct sums -- 4.10. Step 3. Theorem 4.6 -- 4.11. Step 4. Theorem 4.7 -- Chapter 5. Homogeneous directed graphs omitting I[sub(∞)] -- 5.1. A catalog of homogeneous directed graphs -- 5.2. The graph P(3) -- 5.3. The theorem -- 5.4. The major steps in the proof -- 5.5. Proof of the Main Theorem, Part 1 -- 5.6. Proof of the Main Theorem, part 2, n > -- 2 -- 5.7. Case 2.1 of the Main Theorem -- 5.8. Propositions 14 and 15 -- Chapter 6. Propositions 16 to 20 and MT 2.2 -- 6.1. Proposition 16: simple configurations -- 6.2. Proposition 17: induction on n -- 6.3. Proposition 18: extending I[sub(n)] -- 6.4. Proposition 19: ([sup(p)]y,y'). 6.5. Proposition 20: ([sup(p)]y,y[sup(⊥)]) -- 6.6. Toward MT 2.2 -- 6.7. Lemma 6.3 -- 6.8. Lemma 6.4 -- 6.9. Lemma 6.5 -- 6.10. Lemma 6.6 -- Chapter 7. Homogeneous directed graphs embedding I[sub(∞)] -- 7.1. The classification theorem -- 7.2. The main ingredients -- 7.3. Structure of the proof -- 7.4. Steps 4, 6, 7. The Main Theorem -- 7.5. Step 1. Proposition 24: P[sub(3)] -- 7.6. Step 1, Proposition 25: adding L(2) -- 7.7. Step 1, Proposition 26: the operations ± -- 7.8. Step 1, Propositions 27 and 28: some 1-types -- Chapter 8. Theorems 7.6-7.9 -- 8.1. Step 2. Theorems 7.6 and 7.7 -- 8.2. Step 5. Theorem 7.9.T: extending a direct sum -- 8.3. Step 3. Theorem 7.8, 1-types over sums -- 8.4. Theorem 7.8, conclusion -- Appendix: Examples for richer languages -- Bibliography -- Index of Notation -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.
9781470402105
Directed graphs.
Tournaments (Graph theory).
Model theory.
Ramsey theory.
Permutation groups.
Electronic books.
QA166.15 -- .C44 1998eb
510 s;511/.5