100 problems

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

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 $n,t,k$ 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…

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 $\mathcal{H}$ 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…

Deza–Frankl conjecture. For any $k\in\mathbb{N}$ and any $n$ sufficiently large depending on $k$, if $I\subset S_n$ is $k$-intersecting, then

For integers $r\ge 3$ and $r-2\ge t\ge 1$, let $\iota_0(t,n;r)$ denote the maximum size of a $t$-intersecting family in an $r$-partite $r$-uniform hypergraph with all parts of size…

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…

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…

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 $n\ge 2a\ge 2b\ge 4$, and let $A_0\in\binom{[2,n]}{a}$ and $B_0\in\binom{[2,n]}{b}$ satisfy $A_0\cap B_0\ne\emptyset$. Define … … Two families are cross-intersecting if every m…

Frankl–Kiselev–Kupavskii conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ is intersecting and $n>10k$, then

Let $m(k,k)$ denote the maximum product of the sizes of two cross-intersecting families with maximal covering number $k$ in the setting of the paper. The diagonal conjecture. For a…

Let $n$ and $t$ be positive integers, let $cSigma_n$ be the collection of permutations of $[n]$, and let $cA_k$ be the family … A family $\calF\subseteqcSigma_n$ is $t$-intersectin…

Tokushige's conjecture. Let $p_1,p_2\in(0,\frac{1}{t+1})$. If $\mathcal F_1$ and $\mathcal F_2$ are cross $t$-intersecting, then

Complete-intersection conjecture. For any $t$ and $n$, there exists a $j$ such that $\mathcal{F}_{n,t,j}$ is the largest $t$-intersecting set of spanning trees.

Fixed-edge extremal conjecture. If $t\le n/2$, then the largest $t$-intersecting family of spanning trees is

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…

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