Research Interests

I work in a subfield of combinatorics called "Ramsey Theory." I am mostly interested in questions related to "van der Waerden's theorem on arithmetic progressions," which states that "for each k, there exists a (smallest) n = n(k) such that whenever [1, 2, ... , n] is partitioned into two parts, in any way whatsoever, then at least one of these parts must contain a k-term arithmetic progression." (It is known that n(3)=9, n(4)=35, n(5)=178.) For example, one can look for upper and lower bounds for the function n(k) and related functions (allowing partitions into more than two parts, for example). Or, one can replace the set of all k-term arithmetic progressions by a larger or smaller or different set. Or, one can restrict the class of allowable partitions.