Let $\kappa$ be a successor cardinal. We force a universe in which every model of PA of size $\kappa$ extends to a model M of the same size, where M has no splendid extensions. If there is an ineffable cardinal then this statement holds at some cardinal below it, in ZFC.
For any infinite cardinal $\kappa$, we give an example of a centerless group $G$ of cardinality $\kappa$ whose automorphism tower terminates after $\kappa^+$ steps, answering a question of Simon Thomas. The work on this paper was assisted by AI.
Assuming the nonexistence of uncountable measurable cardinals, we construct an $\mathbb F_2$-algebra and a covering class of right modules which is not closed under colimits of countable chains. This gives a counterexample to Enochs'conjecture under $V=L$.
We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardin...
We prove that a relation obtained by composing arbitrary functions of arity $\leq k$ with a relation definable in an $n$-dependent structure is $kn$-dependent. This confirms a conjecture of Chernikov and Hempel. We also demonstrate optimality of the result.
By using an advanced model of AI, it is proved that there exists a commutative ring having a purely-prime ideal which is not purely-maximal. This gives a counterexample to \cite[Conjecture 5.8]{Tarizadeh}.
We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.
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