Infinite homogeneous bipartite graphs with unequal sides

We call a bipartite graph homogeneous if every finite partial automorphism which respects left and right can be extended to a total automorphism. A (κ, λ) bipartite graph is a bipartite graph with left side of size κ and right side of size λ. We show that there is a homogeneous (א₀, 2^א0) bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending on the underlying set theory all homogeneous (א₀, א₁) bipartite graphs may be isomorphic, or there may be 2^א1 many isomorphism types of (א₀, א₁) homogeneous graphs.