16 problems

Let $G$ be a graph of diameter $3$ with $n$ vertices, and let $\Omega_\omega(G)$ denote the $\omega$-cover pebbling parameter used in the paper. For $\omega\geq 1$, diameter-three…

Let $G$ be a nonabelian finite simple group, and let $\operatorname{diam}(G)$ denote the maximum of $\operatorname{diam}(G,X)$ over all generating sets $X$ of $G$. Babai’s conjectu…

Let $G\leq S_n$ be a transitive permutation group. The diameter of $G$ is defined using the maximum over generating sets, namely…

Let $s\geq 3$ and let $\mathcal{D}_{s,n}$ be the $(1:b)$ Maker–Breaker game on the edges of $K_n$ in which Maker wins precisely when her spanning subgraph has diameter at most $s$.…

Let $H_s(n,d)$ denote the maximum diameter of a $d$-dimensional simplicial complex on $n$ vertices. The simplicial-complex diameter conjecture. For every $d\geq 3$, there is a thre…

Conjecture on $m(n,d)$.

DeBiasio–Kamel–McCourt–Sheats conjecture. There exists a constant $d=d(\alpha)$, depending only on $\alpha$, such that every $r$-colouring of $G$ has $(r-1)\alpha$ monochromatic co…

Milićević's conjecture. For every $r$, there is a constant $D(r)$ such that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic components of diameter at m…

English–McCourt–Mattes–Phillips conjecture. In every $2$-coloring of the edges of $G^c$, there exist $A,B\subseteq V$ and colors $i,j\in[2]$ such that

Let $K_n$ be the complete graph on $n$ vertices, let $\omega$ be the edge-disorder environment, and let ${{\mathcal T}}^\omega_{n,\beta_n}$ be the random spanning tree in random en…

Let $G$ be a bridgeless graph of order $n$ and minimum degree $\delta\geq 2$, and let $\overrightarrow{\operatorname{diam}}(G)$ denote the minimum diameter over all strongly connec…

Colorful Bárány–Katchalski–Pach conjecture. Under these hypotheses, there is an index $k\in[2d]$ whose entire family has intersection of diameter at least $c d^{-1/2}$, for some ab…

Non-existence conjecture. There is no bipartite $(d,3,-4)$-graph with $d\ge 6$.

Diameter-three oriented-diameter conjecture. Every bridgeless graph with diameter at most $3$ has oriented diameter at most $9$.

Diameter-three t-pebbling conjecture.

Linear t-pebbling bound conjecture.