18 problems

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.

Let $cmathcal{P}$ be a hereditary property of tournaments, and let $cmathcal{P}_n$ denote the tournaments in $cmathcal{P}$ on $n$ vertices. There exists a constant $calpha>0$ and a…

Algebraicity conjecture. Every $c\in\mathcal{S}$ is either an integer or an irrational algebraic number; equivalently,

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

Exponential–factorial jump conjecture. Either $|\mathcal{P}_n|<c^n$ for some constant $c$, or

Let $\mathcal{P}$ be a hereditary property of ordered hypergraphs. For every constant $c>0$, suppose there exists an $N=N(c)\in\mathbb{N}$ such that $|\mathcal{P}_N|>c^N$. The orde…

Conjecture on hereditary families close to $\mathcal{F}$. For each proper hereditary family $\mathcal{F}$ of graphs there are $C_{\mathcal{F}},d_{\mathcal{F}}>0$ such that if…

Let $\mathcal{H}$ be a non-trivial hereditary graph property, and let accumulation points mean limit points of its edit distance function. Half-interval conjecture. For any non-tri…

Let $\mathcal{H}$ be a non-trivial hereditary graph property, and call a point an accumulation point if it is a limit point of the edit distance function of $\mathcal{H}$. Finitene…

For positive integers $s$ and $t$, let $K_{s,t}$ be the complete bipartite graph with parts of sizes $s$ and $t$, and let $\operatorname{Forb}(K_{s,t})$ denote the hereditary prope…

Let $d4d8$ be a non-trivial hereditary property, and let $d4d2(d4d8)$ be the set of CRGs avoiding its forbidden graphs. For each $p a[0,1]$, let $gK(p)$ denote the quadratic minimi…

Finite CRG representation conjecture. For every nontrivial hereditary property $\mathcal{H}$ and every $\varepsilon>0$, the edit distance function…

Let $\mathcal{L}$ be a finite relational language with relational arity parameter $r_{\mathcal{L}}\geq 2$, and let $\mathcal{H}$ be a fast-growing hereditary $\mathcal{L}$-property…

An ordered graph is a graph whose vertices are linearly ordered, and a hereditary property of ordered graphs is a collection closed under taking ordered induced subgraphs. For a he…

Fix $p_0\in (0,1)$ and let $G(n_0,p_0)$ denote the Erdős–Rényi random graph on $n_0$ vertices with edge-probability $p_0$. Let $\mathcal{H}={\rm Forb}(G(n_0,p_0))$ be the hereditar…