Directed graphs and boron trees

Let L1L1 and L2L2 be two disjoint relational signatures. Let K1K1 and K2K2 be Ramsey classes of rigid relational structures in L1L1 and L2L2 respectively. Let K1⁎K2K1⁎K2 be the class of structures in L1∪L2L1∪L2 whose reducts to L1L1 and L2L2 belong to K1K1 and K2K2 respectively. We give a condition on K1K1 and K2K2 which implies that K1⁎K2K1⁎K2 is a Ramsey class. This is an extension of a result of M. Bodirsky.In the second part of this paper we consider classes OS(2)OS(2), OS(3)OS(3), OBOB and OHOH which are obtained by expanding the class of finite dense local orders, the class of finite circular directed graphs, the class of finite boron tree structures, and the class of rooted trees respectively with linear orderings. We calculate Ramsey degrees for objects in these classes.