 With András Hajnal he proved the Baumgartner–Hajnal theorem, which states that the partition relation formula_4 holds for formula_5 and formula_6.He died in 2011 of a heart attack at his home in Hanover, New Hampshire.The mathematical context in which Baumgartner worked spans Suslin's problem, Ramsey theory, uncountable order types, disjoint refinements, almost disjoint families, cardinal arithmetics, filters, ideals, and partition relations, iterated forcing and Axiom A, proper forcing and the proper forcing axiom, chromatic number of graphs, a thin very-tall superatomic Boolean algebra, closed unbounded sets, and partition relations.