27 problems

Frankl's conjecture. There exists a constant $c$ such that, whenever $k>c d^2$,

Let $S_n$ be the symmetric group, and let $t$-intersecting mean that any two permutations agree on at least $t$ points. For a family $F\subset S_n$, write $|F|$ for its cardinality…

Frankl–Tokushige product conjecture. If $k_i\leq (r-1)n/r$ for every $i\in[r]$, then

Cameron's conjecture. The $t$-umvirate families are the only maximum-sized $t$-intersecting families of permutations. The supplied context says that this conjecture was later prove…

Exact EKR conjecture. Then

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. Elements $x,y\in G_n$ are $t$-space-intersecting when there is a $t$-dimensional subspace $U\leq\mathbb{F}_q^n$ such…

Let $[n]=\{1,\ldots,n\}$, and let $h(n,k,d,s)$ denote the maximum size of a family of $k$-subsets of $[n]$ that contains no $d$ sets with union of size at most $s$ and empty inters…

Ellis's forbidden intersection conjecture. For every fixed $t$ and all $n\geq n_0(t)$, the maximum size of a $(t-1)$-intersection free family $F\subset S_n$ is

Let $\mathcal{F}\subset 2^{[n]}$ be a family of subsets of $[n]$, and let $\ell$ be a positive integer. Suppose that, for some integer $k=k(\ell)$, the intersection of any $k$ dist…

Let $[n]^r$ denote the $r$-partite $r$-uniform hypergraph with vertex classes identified with $[n]$, and let $u(3;)$ denote the maximum number of pairwise disjoint edges. For posi…

Let $X_1,\dots,X_r$ be sets of size $n$, and let $\iota(t,n;r)$ be the maximum size of a $t$-intersecting family $\mathcal{F}\subseteq X_1\times\dots\times X_r$, where two sequence…

Let $L\subseteq [0,k-1]$, and let $m(n,k,L)$ be the maximum size of a family of $k$-subsets of an $n$-element ground set whose pairwise intersection sizes lie in $L$. Suppose…

Let $n,k,\ell$ be positive integers, and let $\mathcal{F}\subseteq\binom{[n]}{k}$. Say that $\mathcal{F}$ avoids intersection $\ell$ when no two members have intersection of size…

Let $kP_3$ denote the disjoint union of $k$ paths on three vertices, and let $r$-EKR mean that a largest intersecting family of independent $r$-sets has the size of a star. Disjoin…

Let $[n]=\{1,\ldots,n\}$, and let $\binom{[n]}{k}$ denote the family of all $k$-subsets of $[n]$. A family $\mathcal{F}\subseteq\binom{[n]}{k}$ is $k$-uniform, and its size is suff…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. Let $Y,Z\subseteq G_n$ be $t$-space-cross-intersecting sets, meaning that every pair in $Y\times Z$ is $t$-space-inte…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. A subset $Y\subseteq G_n$ is $t$-space-intersecting when every pair of elements of $Y$ is $t$-space-intersecting. For…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. For positive integers $t$ and sufficiently large $n$ compared to $t$, let $Y\subseteq G_n$ be a $t$-intersecting set,…

Erdős–Mubayi simplex-cluster conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}$ contains no $d$-cluster, then

Gerbner et al.'s conjecture. Then

Godsil–Meagher's conjecture. A $t$-intersecting version of the Erdős–Ko–Rado theorem should hold: for every $t\in\mathbb{N}$,

Let $\mathcal{L}=\{l_1, l_2, \dots, l_s\}$ be a set of $s$ nonnegative integers with $l_{1}<l_{2}<\dots<l_{s}$, and let $K=\{k_1,k_2,\dots,k_r\}$ be a set of positive integers sati…

Let $r$-uniform set systems be families of $r$-element subsets of $[n]$. For nonnegative integers $a$ and a subset $X\subset[n]$, let $\mathcal{F}_{X,a}$ denote the corresponding e…

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{P}[n]$ denote its power set, and let $d(\mathcal{F})$ be the number of disjoint pairs in a set system $\mathcal{F}\subset\mathcal{P}[n]$. A…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring of dimension $d$. Suppose that … is a complex of finite free modules such that…