A Ramsey phenomenon is an unavoidable phenomenon which happens among a sufficienly large number of objects, and a theorem describing such a phenomenon is called a Ramsey theorem. Since Ramsey, who is a logician, showed a theorem of this style, many theorems have been proved to build up a field in combinatorial set theory. There are various Ramsey theorems, set-theoretical, number-theoretical and ones related to elementary geometry. However, my study is the first one which presents a Ramsey theorem for continuous objects, as knots and spatial graphs.

My Ramsey theorem for spatial graphs states that any embedding of the complete graph on a sufficiently large number of vertices contains a prescribed knot or link. However, this theorem does not hold in general and we need some restriction on embeddings to make the theorem valid. The first restriction was that each edge should be a straight line segment. Further studies have presented Ramsey theorems with weaker restriction and for families of graphs other than the complete graphs.

Miki Miyauchi, now a researched in NTT Laboratory, proved a Ramsey theorem for the compelete bipartite graphs, modifying my method. Her thesis includes it and many other theorems for spatial graphs.

* END * [1999/1/1]