5 problems

Swanepoel's conjecture. The maximum number of edges in a triangle-free penny graph on $n$ vertices is

Let $d(k)$ be the distance associated with the order-$k$ triangular tiling, and let $\alpha$ be the corresponding angle parameter. For $n\geq 1$ and $k\geq 7$, define the displayed…

Let $c(d)$ denote the asymptotic edge-density constant for distance $d$, and let $d(7)$ be the distance associated with the order-$7$ triangular tiling. Strict order-7 optimality c…

For a given positive integer $n$, let $e_d(n)$ be the maximum number of edges among penny graphs with $n$ vertices and distance $d$, and let $d(7)$ be the distance associated with…

Let $e_{d(7)}(n)$ denote the maximum number of edges in a penny graph with $n$ vertices and distance $d(7)$. Exact edge-count conjecture. For all $n obreakgeq 1$, $e{d(7)}(n)$ equa…