182 problems

Let $n$ and $t$ be positive integers with $n>2t$, and let $\mathcal{M}_n(n,n)$ denote the set of perfect matchings of the complete bipartite graph with $n$ vertices in each part, v…

Frankl–Wang's $3/7$ conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting, $n>100k$, and $|\mathcal{F}|>\binom{n-3}{k-3}$, then

Let $[n]=\{1,\ldots,n\}$, and let a family of permutations of $[n]$ be $t$-setwise intersecting if, for any two permutations $\pi$ and $\pi'$ in the family, there exists a $t$-subs…

Ahlswede–Khachatrian conjecture. For all $n$, $k$, and $t$, a largest $t$-intersecting family is isomorphic to one of the families $\mathcal D_i$.

Ellis–Filmus–Friedgut theorem. Every $K_t$-intersecting family of graphs on $[n]$ has size at most

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 $K_{2n}$ be the complete graph on $2n$ vertices, and let $\mathcal{F}$ be a family of perfect matchings of $K_{2n}$. The family is $t$-intersecting if any two of its members sh…

Let $S_n$ be the symmetric group. A family $\mathcal{A} \subset S_n$ is intersecting if for every $\sigma,\pi \in \mathcal{A}$ there is an $i \in [n]$ such that $\sigma(i)=\pi(i)$.…

Frankl–Deza conjecture. Every maximum intersecting family of permutations on $[n]$ is a coset of a point stabilizer.

Let ${\cal F}$ be a family of sets. Call ${\cal F}$ hereditary if every subset of every member of ${\cal F}$ also belongs to ${\cal F}$. Call ${\cal F}$ EKR if some element $x$ sat…

Let $m$ and $k$ be positive integers, let $1\leq t\leq k$, and let $\mathcal{F}\subseteq \left(\binom{[m]}{k}\right)$ be a $t$-intersecting family of $k$-multisets of $[m]$, meanin…

Let $k$ and $ell$ be integers, and let $\mathcal{U}_{k,\ell}$ denote the family of partitions of $[k\ell]$ into $\ell$ blocks of size $k$. A family…

O'Neill–Verstraëte conjecture. Theorem should hold whenever

Frankl–Ota–Tokushige conjecture. There exists a constant $\alpha<1$ such that

Huang's conjecture. If $\mathcal{F}\subset 2^{[n]}$ is intersecting, then

Uniform partial-intersection conjecture. Let $k,c,t$ be positive integers with $t\leq c$ and $n=ck$. If $\mathcal{P}\subseteq\mathcal{U}^n_k$ is partially $t$-intersecting, then ……

Czabarka's conjecture. If $n\leq2k-1$ and $\mathcal{P}\subseteq\mathcal{P}^n_k$ is a partially $2$-intersecting partition system, then …

Complete-intersection conjecture. For $n\geq k\geq t\geq1$, if $\mathcal{P}\subseteq\mathcal{U}^n_k$ is $t$-intersecting, then …

Holroyd and Johnson's conjecture. If $\mathcal{A}\subseteq[n]^{(r)}_k$ is intersecting, then

Let $1\leq t\leq k$, $1\leq r\leq k-1$, and let $p\geq2$ be real. Suppose that … If $\mathcal F\subseteq\binom{[n]}k$ is $t$-intersecting, then, for every $T\in\binom{[n]}t$, defin…

Binary critical-case conjecture. For $q=2$ and $n=2k+1$, the inequality and the complete equality classification in the sharp projective Ore–Hilton–Milner theorem continue to hold.

Extremal-family conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}_{s\text{-stable}}$ is a non-star intersecting family with $\tau(\mathcal{F})\geq 2$, then, for $n$ large enough—s…

Let $n$, $k$, $t$, and $d$ satisfy … Let $\mathcal{F}\subseteq\binom{[n]}{k}$ be $t$-intersecting. Ellis–Keller–Lifshitz stability conjecture. If … then there exists a $t$-subset…

Balogh–Linz–Patkós' conjecture. There exists a positive integer $n_0=n_0(k,t)$ such that if $n+t$ is odd, $n>n_0$, and $\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sper…

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $\mu_{\mathbf p}$ be the product measure on $2^{[n]}$ with coordinate $i$ included independently with probability $p_i$.…