54 problems

Let $R$ be a relation, with domain a non-empty set and with any fixed finite arity. For a structure $R$, let $Sib(R)$ denote the number of isomorphism classes of structures equimor…

A poset $P$ satisfies the finite antichain condition (FAC) if it has no infinite antichain. A partition into antichains is a partition of $P$ whose parts are antichains, and a chai…

Let $(L_i,<_{L_i})$ for $i<\omega$ be an infinite sequence of countable linear orders. A linear order $(L_i,<_{L_i})$ is embeddable into $(L_j,<_{L_j})$ if there is an order-preser…

Aigner's conjectures. The following ordering properties should hold:

Small-order decidability conjecture. The monadic theory of orders of cardinality at most $\aleph_1$ is decidable.

Let $\b5$ be an uncountable regular cardinal, and let ${\emptyset^{\b5}}{\cal H}^{\rho}$ denote the hierarchy introduced in the paper. A poset is strongly $\b5$-scattered if it has…

Let $T$ be a Richardson tableau, and let the order on Richardson components be given together with the operations ${\mathcal T}_{\alpha,\beta}$ and transposition. Richardson-compon…

Let $d\geq 1$, let $\mathcal{H}^d$ be the set of $d$-dimensional shifts ordered by inclusion, and let $\tau^d_{\mathcal{H}}$ be its Hausdorff topology. The shifts of finite type ar…

Let $\mathfrak M:=\{A^{(1)},\ldots,A^{(n)}\}$ be a finite subset of the positive bounded self-adjoint operators $\mathfrak S^{+}$ on a Hilbert space $\mathscr H$, let $S$ be the pa…

Compactness conjecture. Is $Min(Q)$ compact?

Let $\mathcal{L}$ be a locally-finite distributive lattice, and let $\mathcal{P}$ be its set of prime filters. Let $\mathcal{I}$ be the lattice of ideals of $\mathcal{P}$. A lattic…

Splitting-property conjecture. Every countably infinite strongly dense poset has the splitting property.

Let $P$ be a bqo, and consider bqo-preserving operators such as $P\mapsto P^n$ for fixed $n$ and $P\mapsto T(P)$, where $T(P)$ is the Kruskal ordering. The structure of the iterate…

Let $V$ be a finite ground set, and let $\mathcal C$ be an intersection-closed family on $V$ containing $\varnothing$. Suppose that $\mathcal C$ admits a rooted-set representation…

Let $X$ be a chainable continuum, and let an ultrafilter order on $X$ be defined using a sequence of chains obtained from the chainability of $X$. Density conjecture. Such ultrafil…

Let $P$ be a finite poset, let $A$ be an ideal of $P$, and let $h(x)$ denote the expected position of $x$ in a uniformly random linear extension. Write $\max(A)$ for the maximal el…

The regular-cardinal conjecture for sunflowerable dense linear orderings. If $\kappa$ is regular, then every $\kappa$-dense linear ordering is sunflowerable.

Countable-order-type conjecture. $\Pi^{(k)}$ contains subsets of every countable order type.

Structural monotonicity conjecture. Condition (M2) is necessary and sufficient for $\mathit{min}_\mathit{u}$ to fulfill the soundness specification.

Let $F_k^j$ denote the generalized Hofstadter functions indexed by integers $k,j\ge 1$, with pointwise order understood for functions of $n$. Large-index pointwise-order conjecture…

Let $F_k^j$ denote the generalized Hofstadter functions indexed by integers $k,j\ge 1$, with pointwise order understood for functions of $n$. Diagonal pointwise-order conjecture. F…

Let a partially ordered family be a family equipped with a partial order. For two elements, a quasi-supremum is a minimal upper bound and a quasi-minimum is a maximal lower bound.…

Completion conjecture. There are orders with the same monadic theories whose completions do not have the same monadic theories.

Non-isomorphic equivalent-order conjecture. There is an order $M$ such that $M$ and $R$ have the same monadic theory, but $M$ is not isomorphic to $R$.

The Alternate Thomassé conjecture. For every relation $R$ of arbitrary cardinality,