24 problems

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

Let $G$ and $H$ be graphs, and consider the maximum number of copies of $G$ among $n$-vertex graphs $H$ of a prescribed edge density. Nagy's conjecture. For all $G$, this maximum i…

Let $r>1$, let $F$ be a graph with $e(F)>1$, let $m_1,\dots,m_r$ and $\alpha_1,\dots,\alpha_r$ be the parameters defining the symmetric multipartition property…

Let $F$ be a graph with $e(F)>1$, let $m_1,\dots,m_r$ be positive integers, let $\mathcal K$ be the exceptional index set, and let $\alpha_1,\dots,\alpha_r$ be the corresponding pa…

Let $F$ be a graph with $e(F)>1$, let $m\geqslant 2$, and let $\widetilde{\mathcal P}_{1,\dots,1}(F;1/m,\dots,1/m)$ denote the symmetric subgraph-count property for $m$ equal parts…

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…

The Kolmogorov-distance rate conjecture. The same convergence rate as for the distance $d_1$ should hold for the Kolmogorov distance $d_K$, namely

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

Neeman–Radin–Sadun's conjecture. The lower-tail probability satisfies

Gilmer–Kopparty's conjecture. A local central limit theorem should hold for $X_H$.

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 $H$ be a graph with $e(H)\ge 1$. In the Achlioptas process, random choices are replaced by choices made by another player, producing a graph whose appearance and number of copi…

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…

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

Let $\G(n,p)$ be an Erdős–Rényi graph, and let ${\sf H}$ be a fixed $\Delta$-regular connected graph with $v_{\sf H}$ vertices. For two sequences of positive reals, write…

Anticoncentration conjecture.

Upper-tail conjecture. The conclusion of Theorem $H$ remains true under this weaker assumption: if $H$ is nonbipartite, then

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…

Let $H$ be a graph with $e_H\geqslant 1$ edges. Write $X_H$ for the number of copies of $H$ in the random graph, let $\mu_G={\mathbb E}X_G$, let…

Let $n_v$ denote the number of vertices and let $\lambda$ be the parameter governing the expected subgraph counts. In the regime considered, the two Skellam parameters satisfy … Th…