Topological Ramsey Theorem for Complete Bipartite Graphs

An embedding of a graph into a space is linear if each edge is a straight line segment. In 1991, Negami showed that for any given knot, link, or spatial graph there is a sufficiently large complete graph Kn such that every linear embedding of Kn into a space always contains that knot, link, or spatial graph. This paper generalizes this result to cover complete bipartite graphs. The results for complete multipartite graphs and for complete graphs are obtained as corollaries.