25 problems

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite permutation group. A base for $G$ is a sequence of points of $\Omega$ with trivial pointwise stabiliser, and $b(G)$ denotes t…

Golden-rectangle lower-bound conjecture. One might conjecture that

Multiplicative growth conjecture. Numerical evidence suggests that

Let $M$ denote the primitives of rank $4$ for the greedy convolution of length $4$. Let $S$ be the “greedy sifting set”, starting with … and successively augmented by adding the sm…

Let $M$ denote the primitives of rank $4$ for the greedy convolution of length $4$. The rank-one and rank-two primitives are the objects considered below, and divisibility and memb…

In the Gaussian geometric planted matching setting, let the signal-to-noise ratio be $\frac{d}{\log n\,\sigma^2}$, and consider the greedy inner-product algorithm studied in the ci…

Let $\kappa$ denote the target sparsity level, and consider a high dimension-based sparse recovery algorithm using the described greedy approach, with each iteration selecting mult…

Greedy star $L_2$ discrepancy conjecture. Algorithm 1 generates a sequence in $[0,1)$ with the optimal order of star $L_2$ discrepancy.

Greedy star-discrepancy conjecture. The sequence $\mathcal{S}^*$ should be a low-discrepancy sequence, meaning that

Greedy $L_2$ discrepancy conjecture. The sequence $\mathcal{S}^*$ should satisfy

Let the objective function be submodular, let its curvature be the corresponding curvature parameter, and let $\ell$ denote the number of candidate solutions used by the algorithms…

Let $a^{s,(n)}$ be a greedy quantization sequence, and let the regression coefficient compare the optimally dilated greedy sequence with the corresponding greedy sequence. Subseque…

DeVore–Petrova–Wojtaszczyk conjecture. The additional factor of $\sqrt{n}$ in this estimate, although it cannot be removed in general, can be improved for some Banach spaces; in pa…

Steinerberger's no-close-points conjecture. In general, the minimization automatically enforces

Steinerberger's uniform-constant conjecture. The estimate

Steinerberger's small-discrepancy conjecture. The resulting sequence satisfies

Let $G$ be a finite graph. In the greedy coloring game, let $\chi_g^2(G)$ denote the number of colors needed when one round is played, and let $\chi_g^{\infty2}(G)$ denote the numb…

Let $c$ denote the parameter appearing in the assumptions governing the pure greedy algorithm (PGA), and let the convergence rate refer to the decay of the residual sequence under…

Monroe's conjecture. For any distance, lexicographic greedy generation of self-orthogonal codes produces a self-dual code.

Linear dependence conjecture. The term $\mathcal{C}_{|S|-1}$ may be replaced by a linear function of $|S|$.

OGA-FAMMA conjecture. When the variables are preordered by OGA, this minimax optimal rate should be achievable by FAMMA with the factor $2$ replaced by a factor directly proportion…

Probabilistic upper-bound conjecture. These upper bounds hold for all $q$ without any extra conditions. The estimates originate from a conjecture about the unproved steps of a gree…

Greedy-algorithm bounds conjecture. These upper bounds hold for all $q$ without any extra conditions.

Let $H$ be a Hilbert space, and let the Pure Greedy Algorithm (PGA) be the greedy approximation procedure whose iterates are known to converge weakly in $H$. Huber's conjecture. Th…

Let $A=(1,a_1,\ldots,a_k)$ be an orderly currency, and let $P=\{0,1,2,l\}$ be a type-4 set with $4\leq l\leq k$. The sub-currency determined by $P$ is $(1,a_1,a_2,a_l)$. Type-4 sub…