On the order of countable graphs

A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2ω the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (“partnership theorem”). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. s related to universal graphs, rigid graphs (where we solve a problem posed in J. Combin. Theory B 46 (1989) 133) and to the density problem for countable graphs.