51 problems

Let $X$ and $Y$ be finite sets, and let $f:X\times Y\rightarrow \{-1,1\}$ be a Boolean function whose communication matrix has rank $r$. Write $\operatorname{CC}^{\det}(f)$ for its…

Let $\mathbf W^r_{1,\alpha}$ be the weighted Sobolev-type function class, and let $\kappa_m(\mathbf W^r_{1,\alpha})$ denote the minimal cubature error over formulas with $m$ knots.…

For $n\ge 2$, let $\mathcal{B}_n(1)$ be the standard unit closed ball in $\mathbb{R}^n$, and consider the lattice-point discrepancy … as $r\to\infty$. The correct-order conjecture.…

Let $G$ be a nearly linear $k$-uniform hypergraph with $m$ edges, meaning that each pair of vertices lies in at most $O(\log m)$ edges, and let $2\leq r\leq k$. An $r$-cut is a par…

Let $\mathcal{M}=(E,\mathcal{I})$ be a matroid whose ground set $E$ can be partitioned into $k$ disjoint bases, and let $S_1,S_2,\ldots,S_\ell$ be pairwise disjoint subsets of $E$.…

For a random Gaussian matrix $\mathbf{A}\sim\mathcal{N}(0,1)^{n\times m}$ with $m\geq n$, let $\mathrm{SBP}_{\kappa}$ ask for a vector $\mathbf{x}\in\{-1,1\}^m$ satisfying … Here…

Mohr–Pardey–Rautenbach's conjecture. Fix integers $k,n\geq 1$ such that $\binom{n}{2}$ and $(k-1)n/k$ are both even integers. If $\mathbf{c}:E(K_n)\to\{-1,1\}$ is a 0-sum labeling…

Mohr–Pardey–Rautenbach conjecture. There exists a copy $F'$ of $F$ in $K_n$ such that

Matching existence conjecture. If this conjecture holds for $6\leq k\leq 16$, then

Let $f(1),f(2),[?]$ be a sequence with each $f(n)[?][?]$ in $[?]$ and let $x,d[?][?]$ range over $[?]$. Erdős's discrepancy conjecture. For every sequence $[?]$ consisting of $[?]$…

Far-from-Turán discrepancy conjecture. For every $\varepsilon>0$, there exists $c>0$ such that, if $G$ is an $n$-vertex graph that is $\varepsilon$-far from every Turán graph, incl…

Dense regular graph discrepancy conjecture.

Verstraete's conjecture.

Let $T$ be a tree with $\ell$ leaves, let $\mathcal{T}$ denote the family of trees used in the discrepancy definition, and let $\mathcal{D}^1(T,\mathcal{T})$ denote the one-dimensi…

Let $T$ be a tree with $\ell$ leaves, and let $\mathcal{DT}$ be the set of all directed rooted trees, namely trees with a distinguished root and all edges oriented away from it. Wr…

Let $G$ be an $n$-vertex graph, let $\delta(G)$ denote its minimum degree, and let $\mathcal{DHAM}$ be the set of all directed Hamilton cycles. The oriented discrepancy of…

Let $s$ be a de Bruijn sequence over an alphabet of size $q$, and for each letter $a$ let $f_a(i)$ denote its frequency in the initial subsequence of length $i$. Define the discrep…

Extension conjecture. Theorems … should hold for all

Oriented discrepancy conjecture. If $\delta(G)\geq \frac{n+k}{2}$, then the oriented discrepancy of Hamilton cycles in $G$ is at least $k$. This conjecture generalises Dirac's theo…

For $k,n\geq1$, let $\textnormal{HP}_k(n)$ be the $k$-graph defined by the displayed hyperplane-order construction, and let $\mathcal{H}_{\mathrm{HP}_k}$ be its closure under isomo…

Let $\mathcal{L}={R(x,y,z)}$ be the language of ternary $3$-graph relations, and let $\text{XOP}_2$ be a proposed definition given by an infinite scheme of existential sentences. X…

Let $\mathcal{H}_{\mathrm{HP}}$ and $\mathcal{H}_{\mathrm{GS}_p}$ denote the hereditary $3$-graph properties constructed from the hyperplane-order and Green–Sanders examples. Under…

Let $\mathcal{H}$ be a hereditary $3$-graph property. It is weakly stable if there is some $k$ such that every $3$-graph in $\mathcal{H}$ is weakly $k$-stable, where weakly $k$-sta…

Let $\mathcal{H}$ be a hereditary $3$-graph property. For $k\geq 1$, let $\overline{H}(k)$ be the $3$-graph with vertex set ${a_i,b_i,c_i:i\in[k]}$ and edge set…

Let $P_n\subseteq[0,1]^s$ be a point set of size $n$, and let … be the minimum dispersion among such point sets. Bukh's order conjecture. The actual order of the minimum dispersion…