27 problems

Erdős–Ko–Rado conjecture for tilings. For sufficiently large $n$, every intersecting family $\mathcal F\subseteq\mathcal T_n^\Lambda$ satisfies

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

Holroyd–Talbot–Borg conjecture. If $r\leq d/2$, then there is some vertex $v$ of $\Delta$ such that

Let $M_n$ be the graph with vertex set … and edge set … Let $\mathcal{H}^{(p,s)}(n)$ be the family of subsets of $V(M_n)$ that contain exactly $2p+s$ vertices and span exactly $p$…

Minimum-degree conjecture. For every $k>d\geq 0$ and $n\geq 2k+1$, an intersecting family $\mathcal{F}\subset\binom{[n]}{k}$ always satisfies

EKR conjecture. $4\mathcal{H}^{(p,s)}(n)4$ is EKR: every intersecting subfamily has size at most the size of a star, with equality attained by a family consisting of all members co…

Intersection density conjectures. (i) If $|\Omega|$ is a prime power, then $\rho(G)=1$. (ii) If $|\Omega|=2p$, where $p$ is an odd prime, then $1\leq\rho(G)\leq 2$. (iii) If…

Set-wise intersection conjecture. For $k\geq 2t$, the largest set-wise $t$-intersecting family of perfect matchings in $K_{2k}$ has size

Let $q$ be a prime power, and let $\operatorname{PSL}(2,q)$ act on the $2$-subsets of the projective line $\operatorname{PG}(1,q)$. Recall that the group has the Erdős–Ko–Rado (EKR…

Let $G$ range over the groups listed in Table 2 of the source, namely the remaining socles of primitive groups of degree $pq$ that do not admit imprimitive subgroups, where $p$ and…

A $(k,\ell)$-partition is a set partition of $\{1,2,\dots,k\ell\}$ with exactly $\ell$ blocks, each of size $k$. Let $S_{i,j}$ denote the canonical partially 2-intersecting family…

Pendant-path conjecture. The pendant path graph $P_n^*$ is $r$-EKR whenever

For integers $n$ and $k$, let $K_i(n,k)$ denote the random subgraph process of the Kneser graph, and define … … … … Here an EKR graph is one whose maximum independent sets are prec…

Let $r \leq n$, and let $G$ be the vertex-disjoint union of $n$ paths each of length $2$. A family of independent $r$-sets of $G$ is intersecting if any two of its members have a c…

Generalized cluster complex conjecture. Every generalized cluster complex $\mathcal C(m,\Phi)$ is pure-EKR.

Flag pseudo-manifold conjecture. Every flag pseudo-manifold is pure-EKR.

Flag-manifold conjecture. Every flag simplicial manifold is pure-EKR.

Pure-EKR conjecture. Every pure, flag simplicial complex without boundary is pure-EKR.

Kalai's conjecture. Every intersecting family $\mathcal F$ has at most as many elements as there are triangulations of the $(n-1)$-gon.

Let $X_1$ and $X_2$ be disjoint sets with sizes $n_1$ and $n_2$, and let ${X_1, X_2\choose k,\ell}$ denote the family of sets $F$ satisfying $|F\cap X_1|=k$ and $|F\cap X_2|=\ell$.…

Let $G$ be a finite $2$-transitive group. For each coset of a point stabiliser in $G$, let its characteristic vector be the corresponding element of the group algebra $\mathbb{C}G$…

Let ${\cal H}$ be the random $k$-uniform hypergraph with edge-probability $\varphi$, let $\Delta$ denote its maximum degree, and let $\Lambda'(\Delta)$ be the parameter from Theore…

Let $K_{2n}$ be the complete graph on $2n$ vertices, and let $\mathcal{F}$ be a family of its perfect matchings. The family is $t$-intersecting if $|m\cap m'|>t$ for all…

Let $q$ be a prime power, let $k\in\mathbb{N}$, and let $\Gamma=\operatorname{GL}(n,\mathbb{F}_q)$. Define … The graph's independent sets are the $k$-intersecting families of inver…

Let $k,n,t$ be positive integers with $t\leq k$, let $M(n,k)$ be the rectangle representation of $k$-multisets, and let $\mathcal{M}(n,k,t)$ denote the families of $t$-intersecting…