5 problems

Milner–Sauer conjecture. The poset $P$ must contain an antichain of size $\kappa$.

The cofinality conjecture. Precomposition with $F$ preserves marked $(\infty,2)$-colimits if and only if, for every object $d\in\mathbfcal{D}$: (1) there are $c\in\mathbfcal{C}$ an…

Let $(\mathbb{P}, \leq)$ be a poset, and let its cofinality be the singular cardinal $\lambda$. An antichain in $(\mathbb{P}, \leq)$ is a set of pairwise incomparable elements, and…

Consistency conjecture. It is consistent, relative to the existence of large cardinals, that there is $(\kappa,\omega_1)$-reflective forcing where $\kappa$ is the first $\omega_1$-…

Consistency conjecture. It is consistent, relative to the existence of large cardinals, that there is a weakly homogeneous forcing which changes the cofinality of the first inacces…