20 problems

Sharp anticoncentration conjecture. For every tree $T$,

Lee's conjecture. For every tree $T$,

Let $d_0,c>0$ be universal constants. For every integer $d\geq d_0$, let $G$ be a connected $n$-vertex graph with minimum degree at least $d$, and let $\mathcal{T}$ be a uniformly…

Let $\varepsilon>0$ be real. There is a constant $d_{\varepsilon}>0$ such that for every integer $d\geq d_{\varepsilon}$, every connected $n$-vertex $d$-regular graph $G$, and a un…

Vu's conjecture. The probability

Fix $d$, and let $k\in\mathbb{N}$ be a constant depending only on $d$. In the setting of Costello's conjecture, let $f\in\mathbb{F}[x_1,\dots,x_n]$ and let $\xi_1,\dots,\xi_n$ be i…

Fix $d\ge 2$ and $\varepsilon>0$, let $\mathbb{F}\in\\{\mathbb{R},\mathbb{C}\\}$, and let $n$ be sufficiently large in terms of $\varepsilon,d$. Let…

Kwan–Sudakov–Tran strengthened dense inducibility conjecture. There is a constant $C_r$, depending only on $r$, such that

Let $M_n^{sym}$ be a symmetric random matrix whose entries are independent Rademacher variables. Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singul…

Let $0\le d<r$ and $q$ be integers. Let $\mathcal{Z}\subseteq\mathbb{R}^{r}$ be an algebraic variety of dimension $d$ and degree at most $q$. Let…

Let $X_1,\dots,X_n$ be independent random vectors with values in a normed space $\mathcal{S}$. Let $\tilde{\mathcal{Q}}(X,1)$ denote the supremum of the probability that $X$ lies i…

Let $X_1,\dots,X_n$ be independent random vectors in a normed space, and let $\tilde{\mathcal{Q}}(X,1)$ denote the supremum of the probability that $X$ lies in a set of diameter at…

Erdős–McKay conjecture. There is a constant $\delta_C>0$ depending on $C$ such that, for every $C$-Ramsey graph $G$ with $n$ vertices and every integer $x$ satisfying

Conjecture on the optimal bound. The example consisting of groups of repeated coordinates is essentially the worst possible, so that

Anticoncentration conjecture.

Hypergraph logarithm-free inducibility bound conjecture. For any $r,k$ and any $0\le\ell\le\binom{k}{r}$, we have

Logarithm-free inducibility bound conjecture. For all $k$ and all $0\le\ell\le\binom{k}{2}$, we have

Alon–Hefetz–Krivelevich–Tyomkyn's superlinear sparsity conjecture. For all $k,\ell$ satisfying

Consider the two-dimensional quenched-dynamics construction with lattice dimensions $m\times(2m+1)$. Optimal-width conjecture. The required lattice width … is asymptotically optima…

Consider universal random quantum circuits arranged in one spatial dimension, and let the circuit depth be measured as a function of the number $n$ of qubits. Optimal-depth anticon…