14 problems

Füredi's conjecture. For every positive integer $n$, the Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that every chain has size eit…

Let $Q_n$ be the poset of all subsets of $\\{1,\ldots,n\\}$ ordered by inclusion. A chain decomposition partitions $Q_n$ into chains, and two decompositions are orthogonal if every…

Let $B_n$ be the Boolean algebra of all subsets of $[n]$, ordered by inclusion. A chain decomposition of $B_n$ has a type, namely the partition obtained by rearranging its chain si…

Let $F(\beta)$ be a fence and let $L(\beta)$ be its distributive lattice of lower order ideals. For a linear extension of $F(\beta)$, form the corresponding lexicographic chain dec…

Let $F(\beta)$ be a fence and let $L(\beta)$ be its distributive lattice of lower order ideals, with rank sequence governed by the alternatives in Theorem heavy. A saturated chain…

Opposite-property conjecture. The chains $\mathcal{C}_{\mu}$ satisfying these conditions can be chosen to satisfy the opposite property.

The local chain conjecture. There is a set $mathcal{L}$ of local chains of deficit $k$ and an involution mathcal{S}to mathcal{S}^ on $mathcal{L}$ such that: every…

Upper-half dominance conjecture. If $s_1\geq\dots\geq s_M$ is a sequence of positive integers dominated by $\sigma_1',\dots,\sigma_M'$ and

Griggs's conjecture. If $s_1\geq\dots\geq s_M$ is a sequence of positive integers dominated by $\sigma_1,\dots,\sigma_M$ and

A finite ranked poset $P$ is unimodal if its rank sizes form a unimodal sequence, and it has the Normalized Matching Property (NMP) if, for every pair of consecutive ranks, the bip…

Let $Q_n$ be the poset of all subsets of $\\{1,\ldots,n\\}$ ordered by inclusion, and let … A symmetric saturated chain decomposition partitions $Q_n$ into $a_n$ chains, each symme…

Griggs's nesting conjecture. Every normalized matching rank-unimodal poset is nested, meaning that it has a chain decomposition in which, whenever $|C_1|\leq |C_2|$, the set of ran…

Uniform rank-symmetric chain conjecture. For every positive integer $h$ there exists a positive integer $N(h)$ such that, whenever $n>N(h)$ and $n$ is odd, the Boolean lattice…

Rank-symmetric chain conjecture. The Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ rank-symmetric chains such that the size of any two chains di…