A Ramsey Theorem for Graded Lattices

Abstract: The Van der Waerden theorem in Ramsey theory states that if k and r are positive integers and the first N natural numbers are colored using r colors (in any way) then there is a monochromatic arithmetic progression of length k sitting in {1, ..., N}, provided N is sufficiently large. One can interpret this theorem in the language of posets as follows. Let L_n denote the linear poset poset {1, ..., n} with the ordering being the usual ordering of natural numbers. Then the Van der Waerden theorem says that given k and r, if a large enough linear poset is colored with r colors, then one can find a monochromatic 'scaled + translated' copy of L_k in this large poset. It is thus natural to ask if one can prove a similar coloring result for more complicated posets, for instance, the Boolean and set partition posets. In this talk we will discuss a Van der Waerden type theorem in an axiomatic setting of graded lattices and see how one can deduce the Hales-Jewett theorem from it. This is joint work with Amitava Bhattacharya.