9 problems

A well-quasi-order (wqo) is a quasi-order with no infinite descending sequence and no infinite antichain; a better-quasi-order (bqo) is a wqo satisfying the stronger Ramsey-theoret…

Let $\mathcal{C}$ be a hereditary class of finite graphs. Lopez's path-transduction conjecture. If $\mathcal{C}$ is not $2$-well-quasi-ordered, then $\mathcal{C}$ existentially tra…

Let $\mathcal{C}$ be a hereditary class of finite graphs. The bounded clique-width conjecture. If $\mathcal{C}$ is $2$-well-quasi-ordered, then $\mathcal{C}$ has bounded clique-wid…

Let $\operatorname{KT}_{\ell}(\omega)$ denote labelled Kruskal's theorem with labels drawn from $\omega$, and let $\operatorname{KT}_{\ell}(n)$ denote its restriction to labels bel…

Let $(\mathcal{F},\le)$ be a well-quasi-order (wqo), and suppose there are algorithms for basic problems related to $\le$, including deciding whether $X\le Y$ and, given $X$, findi…

Pouzet's conjecture. Every $\omega$-well structured relational structure $\mathcal{A}$ is a reduct of an $\omega$-well structured, totally ordered structure.

Lozin–Razgon–Zamaraev's labelled induced-subgraph conjecture. If $\mathcal{G}$ is defined by a finite set of forbidden induced subgraphs, then $\mathcal{G}$ is well-quasi-ordered b…

Countable-union conjecture. Every well-quasi-ordered poset is a countable union of better quasi-orders.

Let $\mathfrak{F}$ be the partially ordered family of relevant subgroups under embeddability, let $\varepsilon_0$ be the ordinal used to index the chain, and extend the chain by se…