Ramsey-type theorems for spatial graphs and good drawings
A drawing of a graph \(G\) in the plane is said to be good if the following conditions are satisfied: (1) points representing the vertices are all distinct; (2) each edge is represented by a simple arc; (3) any two adjacent edges intersect only at a common end; and (4) any two non-adjacent edges cross in at most one point. A spatial embedding of \(G\) is good if, after applying a suitable ambient isotopy to \(G\), its canonical projection onto the \(xy\)-plane is a good drawing of \(G\). The author has proven the following Ramsey-type theorem: Given a spatial graph \(H\) there exists a pair of natural numbers \((N,M)\) such that any good spatial embedding of \(K_{N,M}\) contains a subgraph which is ambient isotopic to a subdivision of \(H\). This result generalizes earlier results of the author where the role of good drawings was played by rectilinear drawings.
scientific article; zbMATH DE number 1154559