9 problems

Let $d\geq 2$ and let $A\subseteq\mathbb{R}^d$ satisfy $\bar{d}(A)>0$, where $\bar{d}(A)$ denotes its upper limit density. Székely's conjecture. There is a $d_0$ such that every di…

For $d\in\mathbb{N}$, let $B_d(0,X)$ be the closed Euclidean ball of radius $X$ in $\mathbb{R}^d$, and let $N_d(X,\delta)$ be the largest cardinality of a subset of $B_d(0,X)$ for…

Let $B^n$ be the unit ball in $\mathbb{R}^n$, and let $A\subseteq B^n$ be a measurable set containing no pair of points at distance $1$. Write $\mu$ for Lebesgue measure and let…

Let $G(\mathbb{R}^2,\{1\})$ be the graph whose vertices are points of $\mathbb{R}^2$, with two points adjacent when their distance is $1$, and let $\alpha_{\bar{\delta}}$ denote it…

In the normed vector space $(\mathbb{R}^n,\|\cdot\|)$, a measurable set $A\subseteq\mathbb{R}^n$ avoids distance 1 if $\|x-y\|\neq 1$ for all $x,y\in A$. Let…

Erdős's conjecture.

Let $m_1(\mathbb{R}^n)$ denote the maximum density of a subset of $\mathbb{R}^n$ containing no pair of points at distance $1$. Erdős's plane distance-avoiding density conjecture. T…

Let $m_1(\mathbb{R}^2)$ denote the supremum of the upper densities of measurable subsets of the plane containing no pair of points at distance $1$. Erdős's conjecture. … This is a…

Let $m_1(\mathbb{R}^2)$ denote the supremum of the upper densities of measurable subsets of $\mathbb{R}^2$ that avoid distance $1$. Erdős's conjecture. … If true, this would give a…