30 problems

For every pair of linear orders $A$ and $B$ and every $n,m,k,l\in\mathbb{N}_{\geq 1}$, if the ordered sums satisfy $nA+mB\cong kB+lA$, where $nA$ denotes the ordered sum of $n$ cop…

For all finite posets $P$ and $Q$, with $P\times Q$ ordered coordinatewise, $\dim(P\times Q)\ge \dim(P)+\dim(Q)-2$.

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 $P$ be a join-semilattice, and let $\operatorname{Id} P$ denote its lattice of ideals. An ordered set is order-scattered if it does not contain a copy of the order of the ratio…

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…

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

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 $P$ 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.

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.…

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

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 $R$.

Rational-order conjecture. There is a monadic sentence $\psi$ such that

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

Let a DSC be a direct sum of chains, and let $Sib(\mathcal{D})$ denote its sibling number. Suppose … is a countable DSC, where $\mathcal{D}_1$ is bounded, $\mathcal{D}_2$ is a non-…

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…

Let a Specker order and the orders $I,J$ be as described immediately before the statement. Let $T$ be an Aronszajn tree, let $c:T\to 2$, and write $\leq_T$ for its tree order. Shel…

Fish bone conjecture. There exist a chain $C$ and a decomposition $D$ of $V(P)$ into disjoint antichains such that every antichain in $D$ meets $C$.

Let $P=(X,\prec)$ be a finite poset, let $\operatorname{\mathcal{E}}(P)$ be its set of linear extensions, and write $e(P)=|\operatorname{\mathcal{E}}(P)|$. The one-third–two-thirds…

The 1/3–2/3 Conjecture. For any finite poset $P$ which is not a total order, $b(P)\geq\frac{1}{3}$.

The one-third–two-thirds conjecture. For every finite poset $P=(X,\prec)$ that is not a chain, we have

Uniform-poset nonexistence conjecture. There are uniform posets that are not monoid posets.

Level-size characterization. A bi-infinite weak order is a monoid poset if and only if all its levels are of the same size.

Poset extension conjecture. The following conditions are equivalent: