78 problems

Kellerhals–Perren's conjecture. These growth rates should be Perron numbers.

Arratia's conjecture. For every permutation $\beta$ of length $k$,

Let $\mathcal{S}_4$ be the four-star graph, and let $\gamma(\mathcal{S}_4)$ denote its growth parameter. Transcendence conjecture for $\mathcal{S}_4$. The growth parameter…

Perron-number conjecture. The growth rate of any Coxeter system of dimension at most $2$ is a Perron number.

Exponential core growth conjecture. All strongly synchronizing transducers which generate an automaton group of infinite order have exponential core growth rate.

Conjecture on oriented-graph speeds. This predicts a jump from polynomial to at least factorial speed for unlabelled hereditary properties of oriented graphs. The paper poses it as…

Conjecture on the possible bases.

No-accumulation-from-above conjecture. For every $c\in\mathbb{R}$, there exists $\varepsilon=\varepsilon(c)$ such that

Smallest-speed-above-2 conjecture. The smallest $c\in\mathcal{S}$ with $c>2$ is the largest real root of

Exponential growth-rate conjecture. The limit

Let $G$ be a hyperbolic group and let $H$ be any infinite subgroup of $G$. Growth-rate decrease conjecture. The growth rate decreases when taking a quotient of $G$ over $H$. This c…

Let $A_4$ be the Stanley sequence starting from the set $\{0,4\}$. Its growth rate is compared with the scale $n^2/\log n$. The conjecture for $A_4$. The sequence does not have gro…

Polynomial-growth conjecture. The total dimension satisfies

Let $G$ be a graph in which half of the vertices have degree two and the other half have degree three, and let $\rho(G)$ denote its universal covering tree growth rate. Minimum gro…

Amenability conjecture. One has

For a finitely generated group $\Gamma$, let $\operatorname{Exp}(\Gamma) \subset \mathbb{R}$ be the set of exponential growth rates with respect to finite generating sets. For a no…

Let $X$ be a simply-connected space of finite type, and let $\operatorname{rank}_p(\pi_n(X))$ and $\operatorname{rank}_{\mathbb{F}_p}(\operatorname{H}_n(\Omega X;\mathbb{F}_p))$ de…

Let $\pi_n\mathbb{S}$ denote the $n$th stable stem and let $\operatorname{rank}_p$ denote its rank at the prime $p$. Few-differentials conjecture. The stable stems satisfy … This c…

Let $X$ be a finite spectrum of type $h$, and let $\varpi_{\mathrm{T}(h)}(X,n)$ denote the size function measuring its telescopic homotopy groups through degree $n$. Telescopic gro…

Continuous-time irregular growth conjecture. There exist integers $k,d\geq1$ and matrices $A_1,\ldots,A_k\in M_d(\mathbb{R})$ such that, for every unit vector $u\in\mathbb{R}^d$ an…

Let $\mathsf{A}$ be a discrete-time linear switched system whose diagonal blocks $A_k^{(i,i)}$ are fixed, and let the off-diagonal blocks $A^{(i,j)}$ vary freely. Write…

Let threshold languages be repetition-free languages over alphabets whose sizes tend to infinity, and let their growth rates be the exponential growth rates of the languages. Shur–…

Let $G$ be a transitive non-amenable graph with critical and uniqueness thresholds satisfying $p_c<p_u$, and fix a vertex $o$. Let $K_o(\omega_p)$ be the open cluster of $o$ in Ber…

For a real $b2>1$, let $b1'(k,b2)$ be the growth rate of the language of words containing no $b2$-power whose tail has length at most $2$. Stronger tail-restricted growth conjectur…

For any real $b2>1$, let $b1(k,b2)$ denote the growth rate of the language of $b2$-free words over an alphabet of size $k$. Shur's asymptotic growth conjecture. For every fixed int…