Model Theory and Hypergraph Regularity
نویسندگان
چکیده
1. Ultrafilters, ultraproducts and ultralimits 1 1.1. Filters and ultrafilters 1 1.2. Ultralimits 2 1.3. Some model-theoretic notation 3 1.4. Ultraproducts of first-order structures 3 1.5. References 6 2. Graph regularity and measures on ultraproducts 6 2.1. Szemerédi’s regularity lemma 6 2.2. Finitely additive measures 7 2.3. Obtaining countable additivity 9 2.4. Integration for charges (signed f.a. measures) 10 2.5. Measure-theoretic regularity 11 2.6. References 13 3. Hypergraph removal 13 3.1. Removal lemmas 13 3.2. Measure-theoretic hypergraph removal 14 3.3. Szemerédi’s theorem on arithmetic progressions 21 3.4. References 22 4. Regularity lemma for hypergraphs of finite VC-dimension 22 4.1. Bounds in the regularity lemma 22 4.2. VC-dimension 22 4.3. The VC-theorem, ε-approximations and ε-nets 25 4.4. Canonical products of finitely approximable measures 27 4.5. Measure-theoretic regularity for hypergraphs of finite VC-dimension 30 4.6. References 34 5. Regularity lemma for stable hypergraphs 34 References 37
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