12 problems

Let $r\ge 1$ and $s\ge 1$ be integers, and let $H$ be an $r$-uniform hypergraph with $s(H)=s$. Asymptotic equivalence conjecture. Then … The quantity…

Let $r\ge 2$, $s\ge 2$, and $k\ge r+1$ be integers. Let … Let $H$ be an $r$-uniform hypergraph with $s(H)=s$ and $\delta_{s-1}(H)=\delta$. The gamma lower-bound conjecture. Then ……

Let $H$ be a graph, and let $w_H$ be the constant defined by $\operatorname{wsat}(n,H)=(w_H+o(1))n$. Ascoli–He conjecture. Every possible value of $w_H$ is rational. Known results…

Let $s>2$, let $j$ satisfy $2s+j<3s-3$, and let $G$ be an erasable graph associated with the proposed formula for $\mathrm{wsat}(2s+j,K_{s,s})$. The erasable-graph edge conjecture.…

Let $K_{s,s}$ be the balanced complete bipartite graph, let $\mathrm{wsat}(n,K_{s,s})$ denote its weak saturation number, and let $G$ be an erasable graph in the corresponding eras…

Let $K_{s,t}$ be the complete bipartite graph with $2<s\leq t$, and let $j$ satisfy $2\leq j\leq t-3$. Write $\mathrm{wsat}(n,K_{s,t})$ for the weak saturation number of $K_{s,t}$.…

Let $K_s$ be the complete graph on $s$ vertices, and let $\mathrm{wsat}(n,K_s)$ denote the minimum number of edges in an $n$-vertex weakly $K_s$-saturated graph. Bollobás conjectur…

Kalai–Zhukhovich conjecture. For every graph $F$, the equality

Weak-saturation conjecture.

Let $\mathcal{F}=\{F_1,F_2,\ldots\}$ be a family of graphs, and define … for every $H\subset K_n$. Let $M$ be a matroid on $E(K_n)$, and write $M_1\leq M_2$ when every independent…

Let $F$ be a graph, let $G_n\sim G(n,p)$ with fixed $p\in(0,1)$, and let $\mathrm{wsat}(G,F)$ denote the weak saturation number of $F$ in $G$. Write “whp” for “with high probabilit…

For a graph $H$, let $\operatorname{sat}(n,H)$ be its saturation number and let $\operatorname{wsat}(n,H)$ be its weak saturation number. For complete graphs, $K_m$ denotes the com…