Multicolored set multipartite Ramsey numbers

Set multipartite Ramsey numbers were introduced by Burger, Grobler, Stipp and van Vuuren in 2004, generalizing the celebrated Ramsey numbers. In this work we extend set multipartite Ramsey numbers to an arbitrary number of colors. Growth properties, connections with classical Ramsey numbers, general lower and upper bounds are obtained, including some improvements of known bounds. We then focus on the case where a monochromatic bipartite graph is required by exploring density arguments and connections with well-known results.