133 problems

For positive integers $d$, define $h_d^{(a)}(n)$ as the number of partitions of $n$ into parts that are $d$-distinct and at least $a$, and define $\ell_m^{(m_1,m_2)}(n)$ as the num…

Let $\mathcal{S}^{(d)}_n$ be the collection of triangulations of the $d$-sphere, and let $\mu^{(d)}_{n,\beta}$ denote the median of the minimum volume of the Boltzmann random spher…

Let $p\in[0,1)$ and let $\boldsymbol{w}=\boldsymbol{w}(n)\in S_n$ be the random permutation in the non-reduced model. Write $\operatorname{inv}(\boldsymbol{w})$ for its number of i…

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no collection of $k$ edges spanning at most $s$ vertices. For positive intege…

Conjectures for $L_n$ and $M_n$. The following five propositions are conjectured:

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…

Asymptotic growth constant conjecture. The asymptotic growth constant is

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

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

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

Phase transition conjecture. The $\lcm$-density of edge ideals of graphs exhibits a phase transition phenomenon when considering the set of all graphs on $n$ vertices as…

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…

Asymptotic dimensionality conjecture. If $L\ll\sqrt{\operatorname{Var}_{\bm l}}$, then

Layered exponential-growth conjecture. The two maxima have the same exponential growth rate:

Let $d\geq10$ and let $\nu$ be a partition of $d$ with $m_1(\nu)\geq2$. Let $z_\nu$ be the standard centralizer factor, and let $b^X_\nu(\mu^{(1)},\dots,\mu^{(s)},m)$ denote the co…

Let $M$ and $w$ be positive integers with $w\geq M$, and let $K(M,L,w)$ denote the maximum number of codewords in an $M$-channel conflict-avoiding code of length $L$ and weight $w$…

Asymptotic normality conjecture. The Betti numbers of these three quotient spaces are asymptotically normally distributed, and the associated variances grow linearly at the same ra…

Let $S(n)$ denote the maximum number of connected components (clusters) obtainable in a bounded confidence (BC) process with $n$ agents. Half-density limit conjecture. … Determinin…

TAS ubiquity conjecture. A $(\frac 1 2 - o_n(1))$ fraction of all oriented paths of length $n$ are TAS.

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

For $n$ and integers $m,k$ with $2\leqslant k<m<n$, let $f(n)$ be the number of pairs $(m,k)$ for which the prefix reversals $r_n,r_m,r_k$ generate the symmetric group…

Length-four symmetry-count conjecture.

Almost-symmetry-free strings conjecture. For any integer $n\ge4$,

Asymptotic ball Davenport conjecture. For $m\geq\sqrt d$,