4 problems

Failure conjecture. The quotient

Let $\alpha$ be a countable chain. In the relevant obstruction-list theorem, $[\omega]^{<\omega}$ is an additional obstruction precisely when $\alpha$ is not an ordinal; when…

Let $\alpha$ be a chain of size $\kappa$. There is a set $\mathbb{B}$ of at most $2^{\kappa}$ join-semilattices, each with a least element, such that an algebraic lattice $L$ conta…

Let $\alpha$ be an order type, and let $\mathbb{B}_{\alpha}$ be a list of join-semilattices such that an algebraic lattice $L$ contains a chain of order type $I(\alpha)$ if and onl…