Ramsey Numbers for Matroids

Letkandlbe positive integers. The Ramsey numberr(k, l)is the least positive integerrsuch that every graphGwithrvertices contains eitherkmutually adjacent vertices orlmutually non-adjacent vertices. A matroid version of the Ramsey number is defined in this paper. Results which are strikingly similar to the classical graph theorems are obtained. For example, the upper bound of (k+l-2 k-1)forr(k, l)of Erdِs and Szekeres has an analogue in the matroid Ramsey numbers considered here.