Some Ramsey-type results

Abstract

The Ramsey's theorem says that a graph with sufficiently many vertices contains a clique or stable set with many vertices. Now we attach some parameter to every vertex, such as degree. Consider the case a graph with sufficiently many vertices of large degree, we can get the realted Ramsey-type result. The Ramsey's theorem of connected version says that every connected graph with sufficiently many vertices contains an induced path, clique or star with many vertices. Now we require the vertex is non-trivial, i.e. the parameter of this vertex is non-trivial, such as $\operatorname{deg}(v)\ge 2$. A connected graph with sufficiently many non-trivial vertices must contain some special induced subgraph. We also get the non-connected version of this Ramsey-type result as a corollary.