I too have come upon that statement in Higman's paper and wondered which manuscript he is referring to. There are two "papers" of Erdős and Rado that I think it might be.
First, it could be a reference to what became their solution to problem 4358 in the Monthly in 1952.
Second, it could instead be their 1959 paper "A theorem on partial well-ordering of sets of vectors" (J. London Math. Soc. (1959), 222–224).