Abstract:  The infinite Ramsey theorem states that given any coloring of all pairs of natural numbers into two colors, there is an infinite subset of natural numbers in which all pairs have the same color.  When moving from sets to relational structures, some surprising phenomena occur:  The prototypical example is that there is a coloring of pairs of rational numbers into two colors such that both colors persist in any subset of the rationals forming a dense linear order (Sierpínski, 1933).  Likewise for colorings of edges in the Rado graph (Erdös--Hajnal--Pósa, 1975).  The study of optimal bounds for finite colorings of copies (or embeddings) of a finite substructure inside an infinite structure is the subject of big Ramsey degrees.  Optimal bounds are connected with structural expansions which produce analogues of the infinite Ramsey theorem; the pursuit of the optimal structural expansions has led to new connections between logic and structural Ramsey theory.