25 problems

Let $H$ be a fixed graph without isolated vertices. For $0<\beta<1$, let $\mathcal M_H(\beta)$ be the supremum of $t(H,W)$ over graphons $W$ with $t(K_2,W)\le\beta$. For…

Let $W:\Omega^2\to[0,1]$ be a graphon, let $k\ge 2$ be fixed, and let $\mathbb{G}(n,W)$ be the associated inhomogeneous random graph. A $K_{k+1}$-fractional cover of $W$ is a measu…

Let $W:\Omega^2\to[0,1]$ be a graphon, and let $\mathbb{G}(n,W)$ denote the associated inhomogeneous random graph. The four conditions in Proposition are: $W$ is not a connected gr…

Let $\mathbf L$ be a Lebesgue graph, meaning a graph on the standard probability space $[0,1]$ whose edge set is Lebesgue-measurable. A finite graph $F$ has a density in a graph gi…

Partial homogenization conjecture. If the corresponding solutions of are denoted by $(\bm{X}^{(k)}_i(t))_{i\in[N_k]}$ and $(\bm{w}^{(k)}_{i,j}(t))_{i,j\in[N_k]}$, then

Let the assumptions of Theorem hold, with the dynamics of $(\bm{X}_i(t))_{i\in[N]}$ and $(\bm{w}_{i,j}(t))_{i,j\in[N]}$ governed by the SDEs with weight-noise intensity $\eta>0$. L…

The positive graph conjecture. A graph $H$ is positive if and only if it has a stable involution.

Nonattainment conjecture. There exists no $f\in\mathcal F$ such that

Let $H_1$ and $H_2$ be graphs, and let $p\in(0,1)$. A pair $(H_1,H_2)$ is $(p,1-p)$-common when it satisfies the off-diagonal common-pair inequality for every complementary pair of…

Let $W$ be a graphon, meaning a symmetric measurable function $W:[0,1]^2\to[0,1]$, and let $t(H,W)$ denote the homomorphism density of a graph $H$ in $W$. A graph $H$ is strongly c…

Let $W$ be a connected graphon and $K\in\mathbb{N}$. For each $n\in\mathbb{N}$, consider the $W$-edge-incremental process $\mathfrak{X}_n=(G_n^t)_{t=0}^{\binom{n}{2}}$ of order $n$…

Fastest-convergence conjecture. For every such $\mathcal{R}$ and every $\delta\in(0,\frac{1}{2})$,

Let $\mathcal{R}$ be a flip process, let $U$ be a graphon, and let $W$ be a destination of $U$, meaning that $\Phi^{t}U \xrightarrow{\square} W$ as $t\to\infty$. Destination conver…

Apex conjecture. For every positive integer $a$, the graph $H^{+a}$ is common. In particular, every complete tripartite graph $K_{r,s,t}$ is common.

Let $r\ge 3$ and let $W$ be a graphon. For $x,y\in[0,1]$, write $t_{x,y}(K_r^{\bullet\bullet},W)$ for the rooted homomorphism density of the clique $K_r$ with two distinguished roo…

A graphon is a measurable symmetric function $W:[0,1]^2\to[0,1]$. It is finitely forcible if it is uniquely determined, up to weak isomorphism, by the densities of finitely many gr…

Lovász–Szegedy finite-dimensionality conjecture. The space of typical vertices of every finitely forcible graphon has finite dimension.

Let $W$ and $W'$ be equivalent tournament kernels, meaning that they represent the same tournament limit. A measure-preserving transformation is a map from $[0,1]$ to $[0,1]$ prese…

Let $H$ be a graph and $0\leq r\leq 1$. Define $h(x)=x\log x-x+1$ and let $\mathsf{LT}(H,r)$ be the problem of minimizing $\mathbb{E}[h(W)]$ over graphons $W$ subject to…

For $0\leq q\leq p\leq 1$, let $\mathsf{LT}_p(H,q)$ be the problem of minimizing $\mathbb{E}[I_p(W)]$ over graphons $W$ subject to $t(H,W)\leq q^{e(H)}$, where $I_p$ is the rate fu…

Let $G_n$ be a sequence of graphs admitting a graphon limit $\mathcal{G}$, with edge weights following a distribution of pseudo-dimension $q$. Suppose that there exists…

Let $F_1,F_2,\dots,F_n$ be finite graphs, let $t_1,t_2,\dots,t_m$ be real numbers in $[0,1]$, and let $\mathcal{S}$ be the set of graphons $W$ satisfying … A graphon $W$ has polyno…

Let $T(W)$ denote the type space associated with a graphon $W$. Finite-dimensionality conjecture. If $W$ is finitely forcible, then $T(W)$ is finite dimensional. The source intenti…

Let ${\cal W}_0$ be the space of graphons under consideration, and let $T(W)$ denote the type space associated with a graphon $W$. Compactness conjecture. If $W$ is finitely forcib…

Let $F_1,\dots,F_k$ be finite graphs, let $a_1,\dots,a_k$ be prescribed values, and let $t(F_i,W)$ denote the homomorphism density of $F_i$ in a graphon $W$. Finitely forcible solu…