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, 2א0) 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 2א1 many isomorphism types of (א0,א1) homogeneous graphs. Goldstern, Grossberg, Kojman: Bipartite Graphs §0. Introduction A homogeneous graph is one in which every finite partial automorphism extends to a total automorphism. All countable homogeneous graphs were classified in [], and countable tournaments were classified in [] (see also []). When looking at countable homogeneous bipartite graphs, one sees that there are only five types of such graphs: complete bipartite graphs, empty bipartite graphs, perfect matchings, complements of perfect matchings and the countable random bipartite graph. In this paper we study the structure of uncountable homogeneous bipartite graphs which have two sides of unequal cardinalities. We must make the following demand on the notion of automorphism to admit this class of graphs: a bipartite graph has a left and a right side, and automorphisms preserve sides (this is necessary, as otherwise a partial finite automorphisms which switches two vertices from the different sides cannot be extended to a total automorphism). We call a bipartite homogeneous graph with a left side of cardinality κ and a right side of cardinality λ > κ and which is neither complete nor empty, a (κ, λ) saS graph. The name should mean “symmetric asymmetric”, where the symmetry is local, and the asymmetry is global, in having a bigger right hand side. (The demand that saS graphs are neither complete nor empty is to avoid trivial cases). The paper is organized as follows: In Section 1 we classify homogeneous bipartite graphs, and remark that there are only five types of countable homogeneous bipartite graphs. Then we prove the existence of (א0, 2א0) saS graphs. The existence of such graphs answers negatively the following question by J. Kupitz and M. A. Perles: is it true that in every connected locally 3-symmetric (see below) bipartite graph of girth 4 which is not a complete bipartite graphs both sides are of equal cardinality? (Kupitz and Perles proved that the answer is “yes” if the graph is finite). In the second section we count the number of non isomorphic (א0,א1) saS graphs under the assumption of the weak continuum hypothesis. We prove that the weak continuum hypothesis (i.e., 2א0 < 2א1 , which is a consequence of the continuum hypothesis) implies that there are 2א1 pairwise non isomorphic (א0,א1) saS graphs. In the third section we show that ¬CH + MA implies that there is only one (א0,א1) saS graph up to isomorphism. These results together show that the number of isomorphism types of (א0,א1) saS graphs is independent of ZFC, the usual axioms of Set Theory. Our interest in homogeneous bipartite graphs started when M. Perles introduced to us the question of the existence of a locally symmetric infinite bipartite graphs of girth 4 with sides of unequal cardinalities. (See below.) We are grateful to him for this, and not less for his careful reading of the paper and his helpful suggestions. The notation we use is mostly standard, but we nevertheless specify it here. 0.1 NOTATION: (1) A bipartite graph is a triple Γ = 〈L,R,E〉 = 〈LΓ, R, EΓ〉 such that L∩R = ∅, L and R are non-empty and E ⊆ {{x, y} : x ∈ L, y ∈ R}. L∪R is the set of vertices of Γ, E is the set of edges. Members of L and R are called left and right vertices, respectively. Abusing notation, we sometimes write v ∈ Γ, instead of v ∈ L ∪R. Abusing notation even more, we may write L×R for {{x, y} : x ∈ L, y ∈ R}. Γ = 〈L,R,E〉 is a subgraph