Complete Subgraphs of the Coprime Hypergraph of Integers III: Construction

The coprime hypergraph of integers on $n$ vertices $CH{I}_k(n)$ is defined via vertex set $\{1,2,\dots ,n\}$ and hyperedge set $\{\{v_1,v_2,\dots ,v_{k+1}\}\subseteq \{1,2,\dots ,n\}:\gcd (v_1,v_2,\dots ,v_{k+1})=1\}$. In this article we present ideas on how to construct maximal subgraphs in $CH{I}_k(n)$. This continues the author’s earlier work, which dealt with bounds on the size and structural properties of these subgraphs. We succeed in the cases $k\in \{1,2,3\}$ and give promising ideas for $k\ge 4$.