24 problems

Let $B_n$ denote the Boolean lattice of subsets of an $n$-element set, and let a poset be nice when its incomparability graph has the nice property. Boolean-lattice niceness conjec…

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be its power set, and let $G_{\mathcal{F}}$ be the graph whose vertex set is a family $\mathcal{F}\subseteq 2^{[n]}$, with two vertices adja…

Fixed-dimensional Boolean-lattice conjecture. For every fixed $m\in\mathbb{N}$,

Fixed-poset asymptotic conjecture. For $n\in\mathbb{N}$ and fixed $P$,

Weak-versus-induced conjecture. For every $n\in\mathbb{N}$,

Axenovich–author conjecture. For any $\varepsilon>0$, there is a large enough $m_0$ such that for any $m,n\in\mathbb{N}$ with $n\ge m\ge m_0$,

Lu–Thompson conjecture.

Non-modest poset conjecture. There is a fixed poset $P$ such that

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, let $R(Q_n,Q_n)$ be the induced poset Ramsey number, and let $R^{\mathrm{w}}(Q_n,Q_n)$ be the weak poset Rams…

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, and let $R(Q_m,Q_n)$ be the induced poset Ramsey number. Fixed-$m$ asymptotic conjecture. For fixed…

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, let $R(Q_m,Q_n)$ be the induced poset Ramsey number, and let $\varepsilon>0$. Subpolynomial-factor conjecture…

Let $Q_n$ denote the Boolean lattice of subsets of an $n$-element set, and let $R(Q_m,Q_n)$ be the induced poset Ramsey number for the pair $(Q_m,Q_n)$. Lu–Thompson's conjecture. F…

Let $P$ be a poset, and let $\mathop{}\\!\operatorname{extsat}(n,P)$ denote the minimum size of an external $P$-saturated family for the Boolean lattice ${\mathcal B}_n$, as define…

Boolean-lattice asymptotic conjecture. For every fixed $m\in\mathbb{N}$,

Fixed-poset asymptotic conjecture. For every fixed poset $P$,

Let $\boldsymbol{P}_i=\mathbf{V}_{m_i,m_i}$ for $1\leq i\leq k$, where $\mathbf{V}_{m_i,m_i}$ denotes the balanced V-shaped poset with $m_i$ elements on each branch. Write…

Weak Boolean lattice local dimension conjecture.

For integers $k\geq 1$, let $R_k(B_2)$ be the least $N$ such that every coloring of the Boolean lattice $\mathcal{B}_N$ with $k$ colors contains a monochromatic copy of the Boolean…

Let $P$ be a poset, and let $\bar{h}_n(\mathcal{F})$ denote the Lubell function of a family $\mathcal{F}$ of subsets of $[n]$. Define $\lambda_n(P)$ to be the maximum value of the…

Let $Q_m$ and $Q_n$ denote Boolean lattices of dimensions $m$ and $n$, respectively, and let $R(Q_m,Q_n)$ be the least integer $N$ such that every red-blue coloring of the Boolean…

Let $[n]=\{1,2,\ldots,n\}$ and let ${\cal B}_n(m,l)$ be the truncated Boolean lattice consisting of all subsets of $[n]$ of sizes from $m$ through $l$. A cutset is a collection mee…

Minimum-width conjecture. For $n\gg m$,

Let $m=2^n$, and consider families of $m$ sets. A subfamily forms a Boolean algebra of dimension $d$ if it is isomorphic to the Boolean lattice $2^{[d]}$. Boolean-algebra subfamily…

Felsner–Reuter conjecture.