100 problems

Kahn–Kalai conjecture. The threshold satisfies

Let $X$ be a finite nonempty set, let $\mathcal{F}\subseteq 2^X$ be a nontrivial monotone property, and define … and … where $w(G,p)=\sum_{S\in G}p^{|S|}$ and…

For a graph $H$, let $p_c(H)=p_c(n,H)$ be the unique $p$ such that … Define the expectation threshold by … The second Kahn–Kalai conjecture. There is a fixed $K$ such that for any…

Let $F$ be a strictly $1$-balanced graph, and let $p^*$ denote the relevant sharp threshold for the disappearance of $F$-isolated vertices. Johansson–Kahn–Vu threshold conjecture.…

Let $G\sim G(n,n,p)$ be a random bipartite graph with two parts of size $n$, and let $\operatorname{tc}_3(G)$ be the minimum number of monochromatic trees whose vertices cover $G$…

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices in which each $q$-set is present independently with probability $p$. A Steiner system is an…

Let $F$ be a graph on $[n]$, let $e=e(F)$ be its number of edges, and let $p_c(F)$ denote the threshold probability for containing a copy of $F$ in a random graph. Kahn–Narayanan–P…

Let $G_d(n,p)$ be the random geometric graph model, and let $\mathcal{F}$ be a non-trivial monotone graph property, where monotonicity means monotonicity under adding edges. Random…

Optimal-scale conjecture. For each $C>0$ there is an $\varepsilon>0$ such that, for every monotone ${\cal F}$, there exists

Isoperimetric-structure conjecture. Given $C$, there are $K,\delta>0$ such that every $(C\log(1/p),p)$-optimal ${\cal F}$ has both the following properties: there is $R\subseteq X$…

Path-threshold transfer conjecture. There exists a graph sequence ${\cal G}$ such that

Let $k>2$, let $d$ be an integer, and let ${{\mathcal G}_{n,d}}$ be a random $d$-regular graph. Here, $k$-colorable means that the vertices can be colored with $k$ colors so that a…

Let $k>2$, let $c$ be a constant, and let $G(n,m=cn)$ be a random graph. Here, $k$-colorable means that the vertices can be colored with $k$ colors so that adjacent vertices receiv…

Let $H_k(n,m)$ be a uniformly random $k$-uniform hypergraph on $n$ vertices with $m$ hyperedges, and let $c$ denote the hyperedge-to-vertex ratio. A hypergraph is 2-colorable if it…

For a non-empty partial Latin square $P$, define its density by … Let $\mathbf{L}$ be a random Latin square of order $n$. The density threshold conjecture. As $n\to\infty$, … When…

For a graph $H$, let $q_f(H)$ denote its fractional threshold and let … Here $X_F$ is the number of copies of $F$ in $G_{n,p}$. Kahn–Kohayakawa–Qiu conjecture. For every graph $H$,…

Let $r$ be the uniformity, let $H$ be a sufficiently dense $r$-uniform hypergraph, and let $\rho(H)$ denote the density parameter used in the paper. Sharp threshold conjecture. The…

Let $r$ be the uniformity, let $H$ be a sufficiently dense $r$-uniform hypergraph, and let $\rho(H)$ denote the parameter used to describe its density. Sharp threshold conjecture.…

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices, and let an $(n,q,r)$-Steiner system be a collection of $q$-sets containing every $r$-set exactly…

Let $\mathcal{G}^{(q)}(n,p)$ be the random $q$-uniform hypergraph on $n$ vertices, and let an $(n,q,r)$-Steiner system be a collection of $q$-sets containing every $r$-set exactly…

For $q\in[0,1]$, call a graph $J$ $q$-sparse if … where $\mathbb{E}_qX_I$ is the expected number of copies of $I$ in $G_{n,q}$. Sparse-graph reformulation. There is a fixed $K$ suc…

For a graph $H$, let $\mathbb{E}_pX_I$ denote the expected number of copies of $I$ in $G_{n,p}$, and define the fractional expectation threshold by … where $N(H,I)$ is the number o…

Consider the constructed planar quantum low-density parity-check codes, which preserve geometric locality and the LDPC structure. Let the (pseudo)-threshold denote the noise thresh…

Let $d\geq 3$ and let $F=F(n)$ be a sequence of $d$-regular graphs on $[n]$, with $n\in\mathbb{N}$. For a subgraph $\tilde F\subset F$, let $\partial_e(\tilde F)$ denote its edge b…

Let $G(n,p)$ be the binomial random graph, and let a square of a Hamilton cycle be the graph obtained from a Hamilton cycle by adding an edge between every pair of vertices whose d…