7 problems

Let ${\cal G}=\{G_1,\ldots,G_n,\ldots\}$ be a graph sequence, and let $th({\cal G})$ denote its threshold when one exists. Threshold existence conjecture. Every graph sequence…

Let $t_1$ and $t_2$ be functions such that $t_1\in\Omega(n^{1/2})$, $t_2\in O(n)$, and $t_1\ll t_2$. Threshold-range conjecture. There is a graph sequence…

Bollobás–Thomason threshold conjecture. The property $\mathcal{P}$ has a threshold in $\mathbb{G}(n,p\,; \,\binom{[n]}{k})$.

A threshold function is a map $f:\Sigma^N\to\Sigma$, where $\Sigma$ is the binary state space. A function is strongly neutralizable when it has the property that its associated Isi…

Let $f535^n$ denote the Boolean cube, let $f539_n$ be the class of threshold Boolean functions on $n$ variables, and let $f539_{f539_n}(f)$ denote the specification number of $f$ w…

Anthony–Brightwell–Shawe-Taylor conjecture. If $f$ has specification number $n+1$, then $f$ is nested:

Let $H$ be a graph with a color-critical edge $e$, meaning that $\chi(H-e)<\chi(H)$. For a graph $G$, let $t_H(G)$ and $b_H(G)$ be the maximum values of $|K|$ over, respectively,…