Mathematical Sciences: Formal Systems and Combinatorics

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Timothy Carlson's work in finite and infinite combinatorics provides powerful unifying generalizations of various so-called Ramsey theorems. The simplest Ramsey theorem is the following: if six points in space are connected in all possible ways by straight line segments and each segment is painted either red or green, then there will always be an entire triangle whose sides are of the same color. This theorem has been generalized beyond easy recognition in a variety of different ways, but all such generalizations are referred to as Ramsey theorems. They have a host of significant consequences in geometry and number theory and elsewhere, and it is very useful to have unifying principles which provide economical derivations for these corollaries. Further results of this nature will be sought.