129 problems

For every nonnegative integer $t$, every integer $m\ge 2$, and every family of pairs of finite sets $(A_i,B_i)_{i=1}^m$ satisfying $|A_i\cap B_i|\le t$ for all $i$ and…

Fix positive integers $N,k,n$ with $n\ge k$. Let $\mathfrak F_{N,k}$ be the class of finite union-closed families $\mathcal F$ of finite sets satisfying…

For every sequence of families $\mathcal{A}_n\subseteq 2^{[n]}$ such that each $\mathcal{A}_n$ is increasing, regular, and $3$-wise intersecting, prove that…

Let $\mathrm{Sym}(n,k)$ be the set of permutations of $[n]=\{1,\ldots,n\}$ having exactly $k$ cycles. A family $\mathcal{F}\subseteq\mathrm{Sym}(n,k)$ is intersecting if, for every…

Let $[n]=\{1,\ldots,n\}$, let ${\mathcal A}\subseteq 2^{[n]}$, and suppose ${\mathcal A}$ satisfies Condition 1: there is a filter ${\mathcal F}\subseteq 2^{[n]}$ and a bijection…

Chvátal's conjecture. Every subset-closed family of sets is ${\sf EKR}$.

Let $G$ be a graph, let $calI^{(r)}(G)$ denote the family of independent $r$-subsets of $V(G)$, and let $calI^{(r)}_v(G)$ denote the star centered at $v\in V(G)$. Write $cmu(G)$ fo…

Let $P$ be a finite poset. Write … and … Define $\pi(P)=\lim_{n\to\infty}La(n,P)/\binom{n}{\lfloor n/2\rfloor}$ and…

Frankl's conjecture. For every IU-family $\mathcal F\subseteq 2^{[n]}$,

Let $P$ be a finite poset. For a family $\mathcal F\subseteq 2^{[n]}$, let $La(n,P)$ be the largest size of a $P$-free family, and let $La^*(n,P)$ be the largest size of an induced…

O'Neill's supersaturation conjecture. If $\mathcal{A}\subset 2^{[n]}$ consists of even-sized subsets and

Friedgut–Kahn–Kalai–Keller antipodal correlation conjecture. For every increasing $f,g:\{0,1\}^n\to\{0,1\}$, if $g$ is antipodal, then

For positive integers $I$ and $J$, an $(I,J,n)$-DGR is a collection of $I$ disjoint Golomb rulers, each a $J$-subset of $\{1,2,\ldots,n\}$. Let $H(I,J)$ be the least positive integ…

Let $[n]=\{1,\dots,n\}$, let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$, and let a $d$-simplex be a collection of $d+1$ members of $\binom{[n]}{k}$ whose total inte…

Let $\mathcal{F}$ be a finite family of sets that is union-closed, meaning that the union of any two members belongs to $\mathcal{F}$. Suppose that every set in $\mathcal{F}$ has s…

Lexicographic initial-segment conjecture. For fixed $n>k\geq3$, if $[S_m]$ is an FC-family for some positive integer $m$ and has universe size $n$, then

Erdős–Chvátal simplex conjecture. Every family $\mathcal{F}\subseteq\binom{[n]}{k}$ without an $r$-simplex contains at most

Let $n,t,k$ be positive integers, and let ${\mathcal B}(t,k)$ denote the family constructed in the source: it contains the $(n+t-1)/2$-subsets containing $[1,t]$, all sets whose si…

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$, and let $\binom{[n]}{k}$ be the family of $k$-subsets of $[n]$. Let $\mathcal F=\{F_1,\ldots,F_t\}$ be a family of subsets o…

Let $\mathcal{H}$ be a hereditary family, with $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ its $r$th and $s$th levels. For cross-intersecting subfamilies, let…

Let $\mathcal{H}$ be a hereditary family, let $\mu(\mathcal{H})$ denote the size of a smallest base of $\mathcal{H}$, and let $\mathcal{H}^{(r)}$ be its $r$th level. The $r$th leve…

Cross-intersection conjecture for hereditary families. If $\mu(\mathcal{H})\geq 2r$ and $\mathcal{A},\mathcal{B}\subseteq\mathcal{H}^{(r)}$ are cross-intersecting, then

Let $t\leq r$ be two natural numbers. A family of faces is $t$-intersecting if every pair of faces in it has intersection of cardinality at least $t$. Let $\Delta$ be a simplicial…

Asymptotic product conjecture. For fixed $k\geq 2$ and sufficiently large $n$ with respect to $k$,

For each $k$, let $m_{\ell}(k)$ denote the extremal quantity corresponding to the $\ell$-diversity introduced in the paper. Asymptotic vanishing-ratio conjecture. For all…