The edge-density conjecture for intrinsic knotting of bipartite graphs

Let $G$ be a bipartite graph with at least five vertices in each part. Write $v(G)=|V(G)|$ for its number of vertices and $e(G)=|E(G)|$ for its number of edges. Intrinsic-knotting conjecture. If

$$e(G) \geq 4v(G)-17,$$

then $G$ is intrinsically knotted, meaning that every tame embedding of $G$ in $\mathbb{R}^3$ contains a nontrivially knotted cycle. The paper proves the conjecture when one part has exactly five or exactly six vertices, but the general case remains open; it also establishes an asymptotic version with a part-size-dependent additive constant.