11 problems

Robust reconstruction independence conjecture. The robust reconstruction accuracy is independent of $\delta$ for any parameters, and the same conclusion holds for more general tree…

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let $k<\frac{n}{4}$. M…

Square-root saturation exponent conjecture. Away from the critical window, the exponent $\ell/3$ in the below-threshold bound can be improved to $\ell/2$, but not beyond; equivalen…

Exact-BP dichotomy conjecture. Exact BP should obey the same Kesten–Stigum dichotomy as the pairwise rule, with, below the threshold, a strictly faster saturation exponent than the…

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let…

Let $k\in\mathbb{N}_{\geq 3}$, let $X$ be a taxon set, and let $T$ and $\widetilde{T}$ be binary phylogenetic trees on $X$. For a binary phylogenetic tree $T$, let $A_k(T)$ denote…

Let $T$ be a binary phylogenetic tree with $n$ leaves, and let $A_k(T)$ denote the set of binary characters requiring precisely $k$ nucleotide substitutions on $T$. A tree is a max…

Let $T$ be a tree with a finite set of leaves, and choose a sequence of leaves at random according to the sampling scheme in the paper. Let $W_k$ denote the length of the minimal s…

Consider the noisy tree broadcast process and robust reconstruction accuracy described above, for regular trees and Galton–Watson trees with Poisson offspring. General-tree and low…

Optimality conjecture. No distance-based algorithm has $l_\infty$-radius greater than $\frac{1}{4}$ for general trees. Consequently, if this assertion holds, the RNJ algorithm achi…

Projection-cone bound conjecture. Every maximal cell in $\mathcal{Q}_n$ is obtained as the intersection of at most $(n-1)!$ projection cones.