12 problems

Let $K_m$ be the complete graph on $m$ vertices, let $\mathcal{T}$ be its uniform spanning tree, and let $d_{\mathcal{T}}(x,y)$ denote the distance between vertices $x$ and $y$ in…

Odd-torus independent-set conjecture. Both displayed asymptotic formulas hold as $n\to\infty$.

Asymptotic tightness conjecture. The bound above is asymptotically tight.

Let $H_n(s)$ be the function defined from the discrete-torus spectral zeta function by the asymptotic expansion … where … Here $\zeta(\Delta,s)$ is the spectral zeta function of th…

Long's conjecture. The graph $G$ contains a path of length at least

Let $\mathrm{DT}^{(d)}_m$ be the discrete $d$-dimensional torus, and let $N^{(d)}_{m,h}(n;\mu)$ denote the number of reduced paths modulo $m$ in $\mathbb{Z}^d$ starting at the orig…

The quantities $X^{(d)}_{m,h}(n;\mu)$ are defined by the explicit formula preceding these observations, for the parameters and partition $\mu$ used there. Integrality conjecture. I…

Let $f=(f_1,\ldots,f_m):\mathbb{Z}_k^n\to\mathbb{Z}_l^m$ be $\alpha$-Lipschitz with respect to the $L^1$ norm. Coordinatewise junta conjecture. For every $\delta,\epsilon>0$, at le…

Let $f:\mathbb{Z}_k^n\to\mathbb{Z}_l$ satisfy … Here $|\cdot|'$ is the source's normalized difference notation. Junta-size conjecture. The function $f$ is $\epsilon$-close to an…

Engelberg–Gleichsner threshold conjecture. The threshold satisfies

Rigidity conjecture. The rigidity phenomenon described by the main theorem has an analog for proper colorings with more than $3$ colors.

Let the cover time of a finite graph be the time required for a random walk to visit every vertex. For the two-dimensional discrete torus, consider the cover time under the suitabl…