15 problems

Let $F$ be a graph with $e(F)>1$, and let $m=|F|$. For a graph property $\mathcal P(F;1/m,\dots,1/m)$ and its weaker version \widetilde\mathcal P(F;1/m,\dots,1/m), the source con…

Alon's star-forest extremal-graph conjecture. For every integer $m>0$, or at least for all sufficiently large $m$, there exists a star forest $G_m$ with $e(G_m)=m$ such that

Alon's asymptotic subgraph-count conjecture. There is a positive constant $b(H)$ such that

Let $H$) be a fixed graph containing $t$ triangles and $s$ two stars. Under the same conditions as in the two-star conditional central limit theorem, let $F_n$ denote the number of…

Let $H$ be a strictly balanced graph, with $v_H$ vertices, $e_H$ edges, maximum degree $\Delta$, and parameter $\alpha_H^*$. Let $\Psi_{p,H}(\delta)$ denote the mean-field variatio…

Influential-pair obstruction conjecture. If for every $N$ there is some $n>N$ such that $G_n$ has an influential pair of vertices ${\mathbf{w}}=(v,w)\in V(G_n)_2$, then

Triangle-free graph discontinuity conjecture. The function

Let $H$ be a graph with $e(H)\ge 1$, and let $m(H)$ denote its maximum density over nonempty subgraphs, namely … Consider the $b$-Waiter–Client game on the edges of $K_n$, where Wa…

Let $s$ and $t$ be positive integers with $s\leq t$, and let $\mathcal G$ be the class of graphs containing no $K_{s,t}$ minor. Let $H$ be a graph containing no $K_{s,t}$ minor. Fo…

Let $d$ be a nonnegative integer, let $\mathcal D_d$ be the class of $d$-degenerate graphs, and let $H$ be a $d$-degenerate graph. For a graph $H$, let $\operatorname{flap}_d(H)$ d…

Let $\mathcal F$ be a finite set of graphs, let $H$ be a graph, and let $\mathcal G$ be the class of all planar graphs with no subgraph isomorphic to any member of $\mathcal F$. As…

Integer-power conjecture. For all finite sets of graphs $\mathcal{F}$ and all graphs $H$, there is an integer $k$ such that

Eppstein's conjecture. For every graph $H$, there exists a non-negative integer $k$ such that

Anticoncentration conjecture.

Let $H$ be a graph, let $m_H$ denote its appearance-threshold parameter, and let the DeMarco–Kahn upper tail conjecture be the assertion that the upper-tail estimate stated above h…