228 problems

Let $n$ be a positive integer and let $k$ satisfy … Consider the number of sparse paving matroids of rank $k$ on an $n$-element ground set and the total number of matroids on $n$ e…

For $\alpha\in(0,1)$, let $N_{\alpha}(d)$ be the maximum number of equiangular lines in $\mathbb{R}^d$ with fixed angle $\arccos\alpha$. Bukh's conjecture. For each relevant positi…

Charpin's conjecture. Almost all cyclic codes are standard.

Exponential decay conjecture. Both proportions exponentially decay with respect to crossing number:

Non-modest poset conjecture. There is a fixed poset $P$ such that

Let $\mathcal D_n$ be the average number of iterations of the ordinary stack-sorting map $\operatorname{\mathtt{s}}$ needed to send an element of $S_n$ to the identity permutation,…

Upper-bound conjecture. The values $d_n$ are bounded from above by $0.6040$.

Let $\lambda$ be a balanced Young diagram and let $\pi,\pi_1,\dots,\pi_l$ be permutations, with the notation of the approximate factorization estimates for characters. The leading-…

Stabilized-difference conjecture. Recall the integers $c_{n,2}^\bullet(k)$ introduced in Appendix B. We conjecture that, for each fixed $k\ge0$,

Let $k, ell\ge 0$ and $\beta,z>0$. Define … and let $r_{(k,\ell),\beta}(z)$ be the corresponding limiting integral ratio. Second-term oscillation conjecture. As $n\to\infty$, the q…

The leading-amplitude conjecture. The leading logarithmic amplitude is

Asymptotic growth constant conjecture. The asymptotic growth constant is

Let $g\geq 0$ and $p\geq 3$ be fixed integers, and let $\boldsymbol{\alpha}=(\alpha_1,\alpha_2,\ldots,\alpha_p)$ satisfy $\alpha_k>0$ and $\sum_{k=1}^p\alpha_k=1$. Define … For…

Fixed-degree defect-hierarchy conjecture. For every $k\geqslant0$ and every $d\geqslant2k+3$, the recursive orbit weights of levels at most $k$ are nonnegative and form a fractiona…

For positive integers $\Delta$ and $t$, let $h_t(\Delta)$ be the smallest integer such that every graph $G$ with at least $h_t(\Delta)$ edges and maximum degree…

Monotonicity conjecture. The sequence

West's convergence conjecture. The quotient $\mathcal{D}_n/n$ converges as $n$ approaches infinity.

Asymptotic counting conjecture. For each prime power $q$, we have

Asymptotic counting conjecture. For all $r\geq 2$ and prime powers $q$, we have

Asymptotic excess conjecture. As $d\to\infty$,

Let $m(n)$ be the maximum sort-number among permutations of length $n$, where the sort-number is the number of iterations of $SC_{231}$ needed to reach the relevant sorted class. M…

Let $a(n)$ be the average sort-number of permutations of length $n$. Power-law exponent conjecture. Find the value of $k$ such that … is finite, or prove that no such value exists.…

Let $a(n)$ be the average sort-number of permutations of length $n$, where the sort-number is the number of iterations of $SC_{231}$ needed to reach the relevant sorted class. Aver…

Let $W_{2n}$ be the wheel on $2n+1$ vertices, and let $R(W_{2n})=R(W_{2n},W_{2n})$ denote the diagonal Ramsey number for two copies of this wheel. Six-times-linear wheel conjecture…

Record-sparse regime.