Self-complementary Graphs and Weak Consequences of the Axiom of Choice

In ZF the existence of self-complementary graphs on every infinite set is equivalent to the statement that every infinite set is ’even’. We prove a generalization to k-uniform t-complementary hypergraphs. 1. Self-complementary graphs A graph G is called self-complementary if it is isomorphic to its complement G, which has the same vertices as G and exactly those edges that are not in G. First we define some generalizations thereof. Let [V ]k denote the set of k-element subsets of V . A k-uniform hypergraph (k ∈ N = {1, 2, . . . }) is a set of vertices V together with a set of k-edges E ⊆ [V ]k. For a k-edge e = {x1, . . . , xk} and a permutation f of V let fe = {fx1, . . . , fxk}. For a set of k-edges E let fE = {fe : e ∈ E}. A k-uniform t-complemented hypergraph (t ∈ N), abbreviated (k, t)−cg, is a triple (V,E, f) consisting of a set of vertices V , a set of k-edges E ⊆ [V ]k and a permutation f of V such that, with ] denoting disjoint union, [V ]k = E ] fE ] fE ] · · · ] f t−1E. A k-uniform ω-complemented hypergraph, abbreviated (k, ω)−cg, is a triple (V,E, f) consisting of a set of vertices V , a set of k-edges E ⊆ [V ]k and a permutation f of V such that