29 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…

Let $S_{2,1}$ be the relevant colored graph, and let an $S_{2,1}$-good sequence $(G_n)$ have limiting edge density $\beta$ and profile $\rho(S_{2,1},(G_n))$. Define the constructio…

Semi-inducibility conjecture for $H_3$. The semi-induced density satisfies

Strong modeling limit conjecture. Every FO-convergent sequence of graphs in $\mathscr C$ has a strong modeling FO-limit. The paper proves existence of modeling FO-limits for monadi…

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…

Nešetřil–Ossona de Mendez's strong mass-transport conjecture. Every first-order convergent sequence of graphs from ${\cal C}$ has a modeling limit that satisfies the strong finitar…

For $2\leq i\leq k$, let $a_i$ be coefficients and define … where $p(K_i,G)$ is the induced density of $K_i$. Let $OPT$ denote the set of optimal vectors, and write $x_0$ for the c…

For $2\leq i\leq k$, let $a_i$ be coefficients and define a graph parameter by … where $p(K_i,G)$ is the induced density of the clique $K_i$. Let $OPT$ denote the set of optimal ve…

Weighted Aldous–Lyons soficity conjecture. There exists a sequence of $K$-weighted graphs ${G_n,p_n}_{n=1}^\infty$ such that $\alpha$ is the limit of ${G_n,p_n}_{n=1}^\infty$.

Admissible-lift conjecture. Every local-global convergent sequence of graphs in a nowhere dense class has a modeling limit.

Let $q>0$ and let $(\mathbb{G}_n)$ be a sequence of infinite transitive graphs converging locally to an infinite transitive graph $\mathbb{G}$. For an infinite transitive graph…

Constant-community approximation conjecture. There is a constant $c$ independent of $N$, but dependent on the weights $\beta_i$, such that every $X\in\mathcal{X}_f$ is $o(n)$-close…

Finite-graph $\mathrm{INT}_f$-minimizing partition conjecture. For every continuous and strictly concave function $f:[0,1]\rightarrow\mathbb R$ and every $\epsilon>0$, there exist…

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

Let $\mathbf L$ be a modeling. Assume: (i) the theory of $\mathbf L$ has the finite model property; (ii) every interpretation of $\mathbf L$ satisfies the finitary mass transport p…

Let $(G_n)_n$ be a sequence of graphs convergent in the dense model. For a graph $G$ of order $n$, let $\nu_G$ denote its rescaled chromatic measure, defined from the chromatic-roo…

Measurable-function representation conjecture. The limit of every QF-convergent sequence of finite $\sigma$-structures can be represented by a measurable function

Let ${\cal C}$ be a nowhere-dense class of graphs. A sequence of graphs from ${\cal C}$ is first order convergent if the limiting frequency of every first-order property exists. A…

Let $F \neq K_2$ be a non-empty graph with vertex set $\{1,\ldots,m\}$, and let $\alpha_1,\ldots,\alpha_m \in (0,1)$ satisfy … Write $\mathcal{P}(F,\alpha_1,\ldots,\alpha_m)$ for t…

Three-variable kernel conjecture. If $\alpha\notin\{1/3,2/3\}$, then $f=0$ almost everywhere. If $\alpha=1/3$, then, almost everywhere,

For a graph $G$ on $n$ vertices, let $X_n$ be the degree of a uniformly random vertex of $G$. Let $\mathcal{OSG}_n$ and $\mathcal{OSG}^L_n$ denote the classes of unlabeled and labe…

Let $\mathcal{SG}_n$ and $\mathcal{SG}^L_n$ denote, respectively, the classes of unlabeled and labeled string graphs on $n$ vertices, and let $W^4_{1/2}$ be the corresponding graph…

Let $(\mathscr{P}_n)$ be a sequence of branching partitions and let $\widetilde{\mathscr{P}}$ be a branching partition. Write…

Local-global convergence conjecture. A growing sequence of random $d$-regular graphs is local-global convergent with probability one.

Let $\mathbf{F}$ be a vector of graphs, and let $P_{\mathbf{F};\infty}$ be its limit object, a compact convex set whose volume is measured in its ambient finite-dimensional space.…