20 problems

Let $\mathcal C$ be a hereditary class of finite structures, and let $\tilde{\mathcal C}_{\leq \omega}$ be the class of countable structures whose age is included in $\mathcal C$.…

Let $\mathcal{C}$ be a hereditary class of finite graphs. Assume the dichotomy under consideration is the alternative that $\mathcal{C}$ is not $2$-well-quasi-ordered or that its i…

Let $\mathcal{C}$ be a hereditary class of graphs of bounded clique-width. Lopez's correspondence conjecture. The class $\mathcal{C}$ is $2$-well-quasi-ordered if and only if it is…

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 $\mathcal{C}$ be a class of finite structures that is universally well-quasi-ordered. Consider adding a total order on the vertices of every structure in $\mathcal{C}$. Pouzet'…

Let $\mathcal{C}$ be a hereditary class of structures. Being universally well-quasi-ordered means that, for every well-quasi-order of labels, the class of labelled structures from…

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 $S'$ be a finite set of graphs, and let $\operatorname{Forb}_\subseteq(S')$ be the family of graphs containing no member of $S'$ as an induced subgraph. Wqo-bounded-cliquewidth…

Let $S'$ be a finite set of graphs, and let $\operatorname{Forb}_\subseteq(S')$ denote the graphs containing no member of $S'$ as an induced subgraph. Many-forbidden-graphs conject…

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.

Let $\mathsf{LO}$ denote the class of countable linear orders, and let $\operatorname{\preccurlyeq}_{\mathsf{LO}}$ denote embeddability between them. Fraïssé's conjecture. The clas…

A well-quasi-order (wqo) is a quasi-ordered class in which every infinite sequence contains indices $i<j$ with the $i$th element below the $j$th. Let…

Let $\mathcal C$ be an age of finite structures, ordered by embeddability, and let $[\omega]^{<\omega}$ denote the collection of finite subsets of $\omega$, ordered by inclusion. A…

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…

Lozin–Razgon–Zamaraev's conjecture. If a finitely defined hereditary graph class $\mathcal{G}$ is well-quasi-ordered by the induced subgraph relation, then $\mathcal{G}$ has bounde…

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 groups under embeddability, with order relation $\hookrightarrow$. A partial order is a well-quasi-order if it contains no inf…

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…