39 problems

Let $G$ be a graph, and let $\mu(G)$ be the minimum size of a maximal independent set in $G$. For an integer $k$, say that $G$ is $k$-EKR if no intersecting family in the family of…

Let $n\geq 2k$ and let $\mathcal{F},\mathcal{G}\subseteq\binom{[n]}{k}$ be cross $1$-intersecting families. The natural parameter measuring their sizes is…

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…

Minimax independence conjecture. If

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

Let $k$ and $d$ be positive integers with $k>d\ge 2$, and let $\mathcal{F}\subseteq\binom{[n]}{k}$ be an intersecting family. For $S\subseteq[n]$, write…

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

Let $\mathrm{PGL}(2,q)$ act naturally and $2$-transitively on the projective line over the finite field with $q$ elements. For a permutation group $G$ of degree $n$, write…

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…

Extremal self-annihilating subspace conjecture. Equality

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…

Let $X_{t,k,\ell}$ be the graph with vertex set $\mathcal{U}_{k,\ell}$, where two partitions $P$ and $Q$ are adjacent if and only if every pair of blocks $P_i\in P$ and $Q_j\in Q$…

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

Let $K(n,k)$ be the Kneser graph, whose vertices are the $k$-subsets of an $n$-element set, and let a star be the family of all $k$-sets containing a fixed element. In a random ind…

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…

Weak Bruhat lattice EKR conjecture. For all values of $1\le r\le \frac12{n\choose2}$, the set ${\mathscr B}_n(r)$ is EKR. Thus, for any $p\in{\mathscr B}_n(1)$,

Let $\operatorname{KG}_{n,k}(p)$ be the random subgraph of the Kneser graph $\operatorname{KG}_{n,k}$ obtained by retaining each edge independently with probability $p$, and let…

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.