Complete Subgraphs of the Coprime Hypergraph of Integers III: Construction

TL;DR

This paper explores methods to construct maximal complete subgraphs within the coprime hypergraph of integers, advancing understanding for small dimensions and proposing ideas for higher dimensions.

Contribution

It introduces new construction techniques for maximal subgraphs in the coprime hypergraph, extending previous bounds and structural insights for specific cases.

Findings

Constructed maximal subgraphs for k=1,2,3

Proposed promising ideas for k≥4

Extended structural understanding of the hypergraph

Abstract

The coprime hypergraph of integers on $n$ vertices $CHI_k(n)$ 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 $CHI_k(n)$. 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 $k\geq 4$.