CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

By Ramsey’s theorem from 1929, no matter how we color the pairs of natural numbers red and blue, there is an infinite subset $M$ of natural numbers such that all pairs from $M$ are red, or all pairs from $M$ are blue. Moreover, we know that any infinite subset $M$ of the natural numbers under the induced ordering is isomorphic to the natural numbers (as a linear order). However, what we just said does not hold for all infinite linear orders. By Sierpiński’s 1933 result, we know that we can color the pairs of rational numbers red and blue so that no sub-copy of the rational order has only red pairs and no sub-copy has only blue pairs. However, if we used more than $2$ colors, we could always find a sub-copy of the rational order whose pairs take on at most $2$ of the total number of colors. Thus, though we do not have exactly Ramsey’s theorem for the rational order, we do have an instance of a (finite) big Ramsey degree.

In this talk, we will present some recent big Ramsey degree results for internal colorings. The notion of internal coloring will be defined in the talk. This is joint work with Dana Bartošová, Mirna Džamonja and Rehana Patel.