16 problems

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…

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

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

Babai spectrum conjecture for paths. Under these hypotheses,

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 $q\in\mathcal{Q}_n$, 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…

Mossinghoff's symmetry conjecture. The polygon $P_n^*$ has an axis of symmetry corresponding to one particular pendant edge in its diameter graph.

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…

The asymptotic t-broom conjecture. For every integer $k\geq 3$, there exists a function $h=h(k)$ such that, whenever $n\geq h(k)$, the maximum of $e(G_k)$ over all graphs $G$ with…