33 problems

Let $F$ be a fixed graph, let $v(F)$ denote its number of vertices, and let $ind(F)$ denote its inducibility. For a cycle $C_k$ with $k\geq 5$, Pippenger and Golumbic conjectured t…

Let $\mathcal{M}$ be the family of complete multipartite graphs, and let $I_H(x)$ denote the inducibility profile of a graph $H$. Unbounded-local-maxima conjecture. For every $k>0$…

The conjectured profile of $K_{2,2,1}$. The function is conjectured to satisfy

Liu–Pikhurko–Sharifzadeh–Staden conjecture. The inducibility problem is perfectly stable for every complete multipartite graph $F$. Perfect stability would in particular imply that…

Let $K_r(t):=T_r(rt)$ be the complete $r$-partite graph with $r$ parts of size $t$, and let $f(n,K_r(t))=I(K_r(t),n)$ denote the maximum induced density of $K_r(t)$ in an $n$-verte…

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…

Edge inducibility conjecture for odd paths. If $k\geq 5$ is odd, then

Let $C_k$ be a cycle of length $k$, let $c(G,C_k)$ be the number of induced copies of $C_k$ in a graph $G$, and define … A balanced blow-up of $C_k$ replaces each cycle vertex by a…

For an $n$-vertex graph $G$, let $\mathrm{Obj}_{5,5}(G)$ be the number of $5$-vertex subsets inducing exactly $5$ edges, let $\mathrm{Obj}_{5,5}(n)$ be the maximum of this quantity…

Edge-Statistics Conjecture. The edge-inducibility constant satisfies

Let $U_d$ be the configuration in $Q_d$ consisting of one vertex, and let $\lambda(U_d,d)$ denote its $d$-cube density. Ud density conjecture. If $d\geq 5$, then … The right-hand s…

Let $d$ and $i$ be positive integers with $1\leq i<d$. Define … Let $i(K_{i,d-i})$ denote the inducibility of the complete bipartite graph $K_{i,d-i}$. E(d,i) density conjecture. E…

Let $H$ be a configuration in $Q_d$, and let $\lambda(H,d)$ denote its $d$-cube density. Classification conjecture above $5/8$. If … then either $H$ is layered and…

Let $W_7,W_8,W_9,W_{10}$, and $W_{12}$ be the configurations in $Q_3$ listed in Table, and let their lower bounds there be the corresponding values of their 3-cube densities. The e…

Let $C_k$ be the $k$-vertex carousel tournament, and let $C_n$ be the carousel tournament on $n$ vertices. For a tournament $T$, write $I(C_k,T)$ for the number of induced copies o…

Let $S_{1,1}$ be the oriented star with one outgoing edge and one incoming edge at its center, equivalently the directed path on three vertices. Let $i(S_{1,1})$ denote its inducib…

Transitive-tournament-free inducibility conjecture. The number of induced copies of $\vec{P}_k$ over all $\vec{T}_3$-free oriented graphs on $n$ vertices is maximized by a balanced…

Iterated blow-up conjecture. The number of induced copies of $\vec{P}_k$ over all oriented graphs on $n$ vertices is maximized by an iterated balanced blow-up of $\vec{C}_{k+1}$. C…

Sparse-edge decay conjecture. For all $k,\ell\in\mathbb{N}$ with

Hypergraph logarithm-free inducibility bound conjecture. For any $r,k$ and any $0\le\ell\le\binom{k}{r}$, we have

Logarithm-free inducibility bound conjecture. For all $k$ and all $0\le\ell\le\binom{k}{2}$, we have

Alon–Hefetz–Krivelevich–Tyomkyn's superlinear sparsity conjecture. For all $k,\ell$ satisfying

Let $\operatorname{ind}(k,\ell)$ denote the limiting maximum proportion of $k$-vertex subsets inducing exactly $\ell$ edges. Quadratic edge-statistics conjecture. For all pairs…

Let $\operatorname{ind}(k,\ell)$ denote the limiting maximum proportion of $k$-vertex subsets inducing exactly $\ell$ edges. Super-linear edge-statistics conjecture. For all pairs…

For a finite graph $H$, let $\operatorname{ind}(H)$ denote its inducibility, and let $K_{|H|}$ and $\overline{K}_{|H|}$ be respectively the complete and edgeless graphs on $|H|$ ve…