26 problems

Tyomkyn–Uzzell's conjecture. For $k\geq 3$ and $n\geq k+1$, except when $k=3$ and $n=8$, every $n$-vertex graph $G$ such that $G_k$ is triangle-free satisfies

Let $\chi_{\varepsilon}(\mathbb{R}^2)$ denote the least number of colors in a coloring of the plane with no monochromatic pair of points at a distance in…

Let $D$ be a finite set of positive integers, and let $\chi(D)$ denote the chromatic number of the distance graph with distance set $D$. For a fixed integer $k\ge3$, consider the d…

Let $d$ be a dimension and let $D$ be a finite set of distances whose elements are algebraically independent over $\mathbb{Q}$. Let $\chi_{\mathbb{R}^d}(D)$ denote the chromatic nu…

Let $d$ be a dimension, let $D$ be a finite set of forbidden distances, and let $\chi_{\mathbb{R}^d}(D)$ be the chromatic number of the distance graph on $\mathbb{R}^d$ in which pa…

Upper-bound conjecture. Any connected graph $G=(V,E)$ has

Sharp-exponent conjecture. For a range of $1<p_1,\ldots,p_k<\infty$, this estimate holds, and in each case the exponent of $\lambda$ is sharp. The proposed exponent depends only on…

Let $G$ be a finite, simple graph, and let $D_2(G)=G_2$ be its $2$-distance graph: it has the same vertex set as $G$, with two vertices adjacent exactly when their geodesic distanc…

The chromatic number of a distance graph is invariant under scaling the distance, so the Babai number $B_1(\mathbb R^n,\|\cdot\|)$ is the chromatic number at distance $1$. The Eucl…

Let $T$ be the set of positive integers under consideration, and let $G(\mathbb{Q}^3,\sqrt{t})$ denote the graph whose vertices are points of $\mathbb{Q}^3$ with adjacency determin…

Let $T$ be the set of positive integers under consideration in the paper, let $T_1=\{22,58,70,82,142,190,298,330,358,382,478,658,742,862,1222,1978\}$, and let $C_3(t)$ denote the n…

Let $\varepsilon>0$. Consider colorings of the Euclidean plane in which any two points whose distance belongs to $[1-\varepsilon,1+\varepsilon]$ receive different colors. Exoo's co…

Let $q\in\mathcal{Q}_n$, and let $G_0(\operatorname{\mathbb{Q}}^n,q)$ be the connected component containing $\operatorname{\textbf{0}}$ in the distance graph associated with $q$. A…

Let $q_1,q_2\boxtimes$ be quadratic forms on $\operatorname{\mathbb{Q}}^n$, and let $G_0(\operatorname{\mathbb{Q}}^n,q)$ denote the connected component containing…

Realization problem. For any integer $k\ge 7$, there exists $b>1$ such that

Exoo's conjecture. For $b>1$ sufficiently close to $1$, it holds

Let $n\geq 1$ be a number. Let $\Delta_n$ be the graph on $\mathbb{R}^n$ connecting points of rational Euclidean distance. The adjacent distance-graph consistency conjecture. The s…

Let $S\subset\mathbb{R}^3$ be finite, and let $r\colon S\to(0,\infty)$ satisfy $e_r(S)=f_3(\lvert S\rvert)$. Suppose that $\lvert S\rvert$ is sufficiently large. Then there is a se…

Carraher et al.'s odd-even density conjecture. If $k\geqslant 3j$, then

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k\geq8$ with…

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k>6$ with…

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Let $k,\ell\geq 1$. The i…

Let $G(S)$ be the distance graph on the integers with generating set $S$, and let $\overline{\alpha}(S)$ denote its maximum density of an independent set. Fix an odd integer…

The double-broom exception conjecture. The only exceptions to the optimality of the double-broom are $k=2$ and $(n,k)=(7,3)$.

The clique-restricted t-broom conjecture. For every $k\geq 3$ and $t\geq 2$, there is a function $h_2\colon\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ such that, if…