Sh:1065

• Shelah, S., & Ulrich, D. \le_{SP} can have infinitely many classes. Preprint. arXiv: 1804.08523
• Abstract:
  Building off of recent results on Keisler’s order, we show that consistently, \leq_{SP} has infinitely many classes. In particular, we define the property of \leq k-type amalgamation for simple theories, for each 2 \leq k < \omega. If we let T_{n, k} be the theory of the random k-ary, n-clique free random hyper-graph, then T_{n, k} has \leq k-1-type amalgamation but not \leq k-type amalgamation. We show that consistently, if T has \leq k-type amalgamation then T_{k+1, k} \not \leq_{SP} T, thus producing infinitely many \leq_{SP}-classes. The same construction gives a simplified proof of Shelah’s theorem that consistently, the maximal \leq_{SP}-class is exactly the class of unsimple theories. Finally, we show that consistently, if T has <\aleph_0-type amalgamation, then T \leq_{SP} T_{rg}, the theory of the random graph.
• Version 2021-08-26 (20p)

@article{Sh:1065,
 author = {Shelah, Saharon and Ulrich, Danielle},
 title = {{$\le_{SP}$ can have infinitely many classes}},
 note = {\href{https://arxiv.org/abs/1804.08523}{arXiv: 1804.08523}},
 arxiv_number = {1804.08523}
}