Disjoint -amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this article, we show that if a countably categorical theory admits an expansion with disjoint -amalgamation for all , then is pseudofinite. All theories which admit an expansion with disjoint -amalgamation for all are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, and , and show that both are pseudofinite. The theories and are not simple, but they have NSOP. This is established here for for the first time.
References
[1] Ahlman, O., “Simple structures axiomatized by almost sure theories,” Annals of Pure and Applied Logic, vol. 167 (2016), pp. 435–56. 06552723 10.1016/j.apal.2016.02.001[1] Ahlman, O., “Simple structures axiomatized by almost sure theories,” Annals of Pure and Applied Logic, vol. 167 (2016), pp. 435–56. 06552723 10.1016/j.apal.2016.02.001
[3] Baldwin, J. T., M. Koerwien, and M. C. Laskowski, “Disjoint amalgamation in locally finite AEC,” Journal of Symbolic Logic, vol. 82 (2017), pp. 98–119. 06725055 10.1017/jsl.2016.25[3] Baldwin, J. T., M. Koerwien, and M. C. Laskowski, “Disjoint amalgamation in locally finite AEC,” Journal of Symbolic Logic, vol. 82 (2017), pp. 98–119. 06725055 10.1017/jsl.2016.25
[4] Blass, A., G. Exoo, and F. Harary, “Paley graphs satisfy all first-order adjacency axioms,” Journal of Graph Theory, vol. 5 (1981), pp. 435–39. 0472.05058 10.1002/jgt.3190050414[4] Blass, A., G. Exoo, and F. Harary, “Paley graphs satisfy all first-order adjacency axioms,” Journal of Graph Theory, vol. 5 (1981), pp. 435–39. 0472.05058 10.1002/jgt.3190050414
[5] Blass, A., and B. Rossman, “Explicit graphs with extension properties,” Bulletin of the European Association for Theoretical Computer Science EATCS, vol. 86 (2005), pp. 166–75. MR2163472 1169.05373[5] Blass, A., and B. Rossman, “Explicit graphs with extension properties,” Bulletin of the European Association for Theoretical Computer Science EATCS, vol. 86 (2005), pp. 166–75. MR2163472 1169.05373
[6] Brooke-Taylor, A., and D. Testa, “The infinite random simplicial complex,” preprint, arXiv:1308.5517v1 [math.LO]. 1308.5517v1[6] Brooke-Taylor, A., and D. Testa, “The infinite random simplicial complex,” preprint, arXiv:1308.5517v1 [math.LO]. 1308.5517v1
[7] Cameron, P. J., Oligomorphic Permutation Groups, vol. 152 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1990. 0813.20002[7] Cameron, P. J., Oligomorphic Permutation Groups, vol. 152 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1990. 0813.20002
[9] Chatzidakis, Z., and E. Hrushovski, “Model theory of difference fields,” Transactions of the American Mathematical Society, vol. 351 (1999), pp. 2997–3071. 0922.03054 10.1090/S0002-9947-99-02498-8[9] Chatzidakis, Z., and E. Hrushovski, “Model theory of difference fields,” Transactions of the American Mathematical Society, vol. 351 (1999), pp. 2997–3071. 0922.03054 10.1090/S0002-9947-99-02498-8
[10] Cherlin, G., “Combinatorial problems connected with finite homogeneity,” pp. 3–30 in Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), edited by L. A. Bokut, A. I. Kostrikin, and L. Ershov, vol. 131 of Contemporary Mathematics, American Mathematical Society, Providence, 1992.[10] Cherlin, G., “Combinatorial problems connected with finite homogeneity,” pp. 3–30 in Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), edited by L. A. Bokut, A. I. Kostrikin, and L. Ershov, vol. 131 of Contemporary Mathematics, American Mathematical Society, Providence, 1992.
[11] Cherlin, G., “Two problems on homogeneous structures, revisited,” pp. 319–415 in Model Theoretic Methods in Finite Combinatorics, edited by M. Grohe and J. A. Makowsky, vol. 558 of Contemporary Mathematics, American Mathematical Society, Providence, 2011.[11] Cherlin, G., “Two problems on homogeneous structures, revisited,” pp. 319–415 in Model Theoretic Methods in Finite Combinatorics, edited by M. Grohe and J. A. Makowsky, vol. 558 of Contemporary Mathematics, American Mathematical Society, Providence, 2011.
[12] Cherlin, G., and E. Hrushovski, Finite Structures with Few Types, vol. 152 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2003. 1024.03001[12] Cherlin, G., and E. Hrushovski, Finite Structures with Few Types, vol. 152 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2003. 1024.03001
[13] Chernikov, A., and N. Ramsey, “On model-theoretic tree properties,” Journal of Mathematical Logic, vol. 16 (2016). 1402.03043 10.1142/S0219061316500094[13] Chernikov, A., and N. Ramsey, “On model-theoretic tree properties,” Journal of Mathematical Logic, vol. 16 (2016). 1402.03043 10.1142/S0219061316500094
[15] Erdős, P., D. J. Kleitman, and B. L. Rothschild, “Asymptotic enumeration of $K_{n}$-free graphs,” pp. 19–27 in Colloquio Internazionale sulle Teorie Combinatorie, Tomo II (Rome, 1973), vol. 17 of Atti dei Convegni Lincei, Accademia Nazionale dei Lincei, Rome, 1976.[15] Erdős, P., D. J. Kleitman, and B. L. Rothschild, “Asymptotic enumeration of $K_{n}$-free graphs,” pp. 19–27 in Colloquio Internazionale sulle Teorie Combinatorie, Tomo II (Rome, 1973), vol. 17 of Atti dei Convegni Lincei, Accademia Nazionale dei Lincei, Rome, 1976.
[16] Flajolet, P., and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. 1165.05001[16] Flajolet, P., and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. 1165.05001
[17] Goodrick, J., B. Kim, and A. Kolesnikov, “Type-amalgamation properties and polygroupoids in stable theories,” Journal of Mathematical Logic, vol. 15 (2015). 1372.03062 10.1142/S021906131550004X[17] Goodrick, J., B. Kim, and A. Kolesnikov, “Type-amalgamation properties and polygroupoids in stable theories,” Journal of Mathematical Logic, vol. 15 (2015). 1372.03062 10.1142/S021906131550004X
[18] Henson, C. W., “A family of countable homogeneous graphs,” Pacific Journal of Mathematics, vol. 38 (1971), pp. 69–83. 0204.24103 10.2140/pjm.1971.38.69 euclid.pjm/1102970260[18] Henson, C. W., “A family of countable homogeneous graphs,” Pacific Journal of Mathematics, vol. 38 (1971), pp. 69–83. 0204.24103 10.2140/pjm.1971.38.69 euclid.pjm/1102970260
[19] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993. 0789.03031[19] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993. 0789.03031
[20] Hrushovski, E., “Groupoids, imaginaries and internal covers,” Turkish Journal of Mathematics, vol. 36 (2012), pp. 173–98. 1260.03065[20] Hrushovski, E., “Groupoids, imaginaries and internal covers,” Turkish Journal of Mathematics, vol. 36 (2012), pp. 173–98. 1260.03065
[21] Kantor, W. M., M. W. Liebeck, and H. D. Macpherson, “$\aleph_{0}$-categorical structures smoothly approximated by finite substructures,” Proceedings of the London Mathematical Society (3), vol. 59 (1989), pp. 439–63. 0649.03018 10.1112/plms/s3-59.3.439[21] Kantor, W. M., M. W. Liebeck, and H. D. Macpherson, “$\aleph_{0}$-categorical structures smoothly approximated by finite substructures,” Proceedings of the London Mathematical Society (3), vol. 59 (1989), pp. 439–63. 0649.03018 10.1112/plms/s3-59.3.439
[22] Kim, B., and H.-J. Kim, “Notions around tree property 1,” Annals of Pure and Applied Logic, vol. 162 (2011), pp. 698–709. 1241.03039 10.1016/j.apal.2011.02.001[22] Kim, B., and H.-J. Kim, “Notions around tree property 1,” Annals of Pure and Applied Logic, vol. 162 (2011), pp. 698–709. 1241.03039 10.1016/j.apal.2011.02.001
[23] Kim, B., A. S. Kolesnikov, and A. Tsuboi, “Generalized amalgamation and $n$-simplicity,” Annals of Pure and Applied Logic, vol. 155 (2008), pp. 97–114. 1153.03008 10.1016/j.apal.2008.03.005[23] Kim, B., A. S. Kolesnikov, and A. Tsuboi, “Generalized amalgamation and $n$-simplicity,” Annals of Pure and Applied Logic, vol. 155 (2008), pp. 97–114. 1153.03008 10.1016/j.apal.2008.03.005
[24] Kim, B., and A. Pillay, “Simple theories,” Annals of Pure and Applied Logic, vol. 88 (1997), pp. 149–64. 0897.03036 10.1016/S0168-0072(97)00019-5[24] Kim, B., and A. Pillay, “Simple theories,” Annals of Pure and Applied Logic, vol. 88 (1997), pp. 149–64. 0897.03036 10.1016/S0168-0072(97)00019-5
[25] Kim, B., and A. Pillay, “From stability to simplicity,” Bulletin of Symbolic Logic, vol. 4 (1998), pp. 17–36. 0897.03034 10.2307/421004[25] Kim, B., and A. Pillay, “From stability to simplicity,” Bulletin of Symbolic Logic, vol. 4 (1998), pp. 17–36. 0897.03034 10.2307/421004
[27] Macpherson, D., “A survey of homogeneous structures,” Discrete Mathematics, vol. 311 (2011), pp. 1599–1634. MR2800979 1238.03032 10.1016/j.disc.2011.01.024[27] Macpherson, D., “A survey of homogeneous structures,” Discrete Mathematics, vol. 311 (2011), pp. 1599–1634. MR2800979 1238.03032 10.1016/j.disc.2011.01.024
[28] Oberschelp, W., “Monotonicity for structure numbers in theories without identity,” pp. 297–308 in Combinatoire et représentation du groupe symétrique (Strasbourg, 1976), vol. 579 of Lecture Notes in Mathematics, Springer, Berlin, 1977.[28] Oberschelp, W., “Monotonicity for structure numbers in theories without identity,” pp. 297–308 in Combinatoire et représentation du groupe symétrique (Strasbourg, 1976), vol. 579 of Lecture Notes in Mathematics, Springer, Berlin, 1977.
[29] Shelah, S., “Toward classifying unstable theories,” Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229–55. 0874.03043 10.1016/0168-0072(95)00066-6[29] Shelah, S., “Toward classifying unstable theories,” Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229–55. 0874.03043 10.1016/0168-0072(95)00066-6
[30] Thomas, S., “Reducts of random hypergraphs,” Annals of Pure and Applied Logic, vol. 80 (1996), pp. 165–93. MR1402977 0865.03025 10.1016/0168-0072(95)00061-5[30] Thomas, S., “Reducts of random hypergraphs,” Annals of Pure and Applied Logic, vol. 80 (1996), pp. 165–93. MR1402977 0865.03025 10.1016/0168-0072(95)00061-5
