57 problems

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

Let $G$ be a bipartite graph and let $H$ be a graph. The normalized homomorphism density $t^*(G,H)$ is defined by dividing $t(G,H)$ by the appropriate power of the edge density of…

Nešetřil–Ossona de Mendez's modeling-limit conjecture. Every first-order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit.

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, let $m_1,\dots,m_r$ and $\alpha_1,\dots,\alpha_r$ be the parameters defining $\widetilde{\mathcal P}_{m_1,\dots,m_r}(F;\alpha_1,\dots,\alpha_r)$, and let…

Let $F$ be a graph with $m=|V(F)|$ and $e(F)>1$. For $A\subseteq V(F)$, let $e_F(A)$ be the number of edges induced by $A$, let $e_F(A^{\mathsf c})$ be the number induced by its co…

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 $r\geqslant3$, let $m_1,\dots,m_r$ be positive integers with $m=m_1+\cdots+m_r$, and let $\alpha_1,\dots,\alpha_r$ be the associated parameters. An integer $k$ is called bad wh…

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…

Let $m$ be a positive integer, and consider subspaces of $L^2([0,1]^m)$ that are invariant under all measure-preserving bijections of $[0,1]$ onto itself, including subspaces consi…

For each integer $k\geq3$, let $\widetilde{P}_k$ denote the CRG introduced in the paper, and let $p_k$ be the associated parameter defined there. A CRG is $p$-core when it has no a…

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…

For each $Delta a in mathbb{N}$, fix a sequence ${G^Deltai}{i=1}^{infty}$ of $Delta$-regular graphs whose girth tends to infinity. Let $f(G,k)=frac{1}{n}ln|mathscr{C}k(G)|$ be the…

Let $\boldsymbol{G}_{n}$ be the randomly twisted hypercube, and form its twisted cube-connected-cycle by replacing every vertex of $\boldsymbol{G}_{n}$ by an $n$-cycle. The Benjami…

A sequence of partially ordered sets whose homomorphism densities converge has a limit representable by a kernel on an ordered probability space. The standard representation conjec…

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…

Unique inducibility conjecture. If all arities are at most $k$, then

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…

Let $G$ be a graph on $n$ vertices, let $w_G$ be its associated graphon, and let $\widetilde{\Gamma}(G)$ and $\widetilde{\Gamma}(w_G)$ be the graph and graphon parameters defined b…

Let $H$ be a bipartite graph with at least one cycle. In the dense setting, where the edge density $p_n$ is constant, a set of graphs is forcing if convergence of the relevant norm…

Mirror-symmetry characterization conjecture. If $t(F,U)\ge 0$ for every kernel $U$, then $F$ is mirror-symmetric.