192 problems

Beck–Fiala conjecture. Every such matrix satisfies

Erdős's conjecture.

De Bruijn–Erdős conjecture. The quantity $r(\mu_r-1)$ tends to infinity as $r$ increases.

Let $B_2^d$ and $B_\infty^d$ denote the unit balls of the Euclidean and maximum norms in $P{R}^d$, respectively, and let $\mathrm{vb}(B_2^d,B_\infty^d,n)$ be the corresponding vect…

Let $G$ be an oriented graph on $n$ vertices, and let $\delta(G)$ denote its minimum degree. A Hamilton cycle has an edge in one of the two possible directions for each consecutive…

Roth's conjecture. For every such function $f$, the lower bound $D(f)\geqslant cn^{1/4}$ should be improvable to $D(f)\geqslant n^{1/2-o(1)}$. Roth's theorem gives the stated…

The Fibonacci set is a point set associated with the Fibonacci permutation, considered as a construction on the two-dimensional torus $\mathbb{T}^2$. Fibonacci-set conjecture. The…

Let $d$ and $t$ be positive integers, and let $N(d,t)$ denote the minimal number of points in a spherical $t$-design on the unit sphere … A spherical $t$-design is a finite point s…

The subunit-$L^p$ discrepancy conjecture. For every dimension $d\geq2$ and every $0<p<1$, there is a constant depending only on $d$ and $p$ such that

The $L^1$ discrepancy conjecture. For every dimension $d\geq2$, there is a constant depending only on $d$ such that

Square-root discrepancy conjecture. The best bound for this discrepancy, uniformly over all $N$ and all such vector sequences, is of order $\sqrt d$.

Fix an integer $B\geq2$, and let $\mathcal{C}_B$ be the set of irrational numbers whose partial quotients are bounded in average by $B$. For each $n$, consider rational points…

Let $p$ be prime, let $\rho_{a,\tau}$ denote the exponentiation permutation defined using a primitive root $\tau$ modulo $p$, and write $D(\rho_{a,\tau})$ for its discrepancy. The…

Let $p$ be prime, let $\psi_k$ be the arithmetic permutation considered in the paper, and let $D(\psi_k)$ denote its discrepancy. Strong logarithmic discrepancy conjecture. … This…

Let $p$) be prime, let $\mathbb{Z}_p^\times$ denote the nonzero residue classes modulo $p$, and let $\psi_k$ be the arithmetic permutation defined in the surrounding discussion. Wr…

Discrepancy lower-bound conjecture. For all matrices, not only badly approximable ones, the factor $k^{n/2}$ in the denominator of the lower bound in Theorem lowerbound should be r…

Relative discrepancy conjecture. In dimensions $d\geq 5$, the relative error satisfies $\Delta_{\Omega}(t)=O(t^{-2})$ as $t\to\infty$.

Let $d\in\mathbb{N}$, and let an order-2 digital sequence mean a digital sequence over the finite field $\mathbb{F}_2$ with order parameter $2$. For an infinite sequence…

Let $d\geq 2$, and define functions on $[0,1]^d$ by … … and … These functions aggregate only distances between consecutive arguments, unlike the pairwise-distance functions conside…

Let $\mathcal{S}=(\mathcal{I}_1,\mathcal{I}_2,\dots)$ be an infinite-length strategy, and define … Here $\operatorname{disc}(n)$ is the optimal discrepancy for finite strategies of…

For each positive integer $n$, let $\operatorname{disc}(n)$ be the minimum, over all strategies of length $n$ that repeatedly split an existing interval into two starting from…

Let $\operatorname{disc}(n)$ denote the minimum possible discrepancy among strategies producing $n$ intervals, and let the lex-merge strategy be the strategy described in the paper…

Komlós' conjecture. The quantity $\kappa_n$ is bounded by an absolute constant independent of $n$, equivalently $\kappa_n=O(1)$. This is a central conjecture in discrepancy theory.…

Let $n\geq 1$ be an integer, and let $a_1,\ldots,a_n\in\mathbb{C}^n$ satisfy … A complex number of modulus $1$ is unimodular, and a vector in $\mathbb{C}^n$ is a unimodular vector…

Consider the discrepancy problem discussed in the source, for which Spencer's theorem guarantees the existence of a signing with the required discrepancy bounds. Spencer's conjectu…