# Orbit-finite bijections can be smoothed

In his recent post Bartek demonstrated an example of two equivariant sets with atoms that are related by a finitely supported bijection, but not by any equivariant bijection. We show that if the sets are orbit-finite then this situation is impossible.

Proposition. Let and be two orbit-finite equivariant sets with atoms. If there exists a finitely supported bijection then there exists an equivariant bijection .

# Join of two elements

Let be two elements with atoms. The pair  satisfies the following universal property: supports both and , and if is another element with this property, then supports .

We demonstrate the existence of an element denoted , with a dual property: is supported both by and by , and if is another element with this property, then is supported by .  The construction does not require any assumptions on the symmetry.

# Some bijections cannot be smoothed

It can happen that two equivariant sets with atoms are related by a finitely supported bijection, but not by an equivariant bijection.

# Bipartite graphs without bipartite partition

In the equality symmetry, there is a single-orbit graph with the property that  is bipartite in the classical sense (equivalently, has no odd-length cycle) but has no partition into two parts which are finitely supported and not inter-connected. In other words, the graph maps to the single-edge graph via a infinitely-supported homomorphism, but not via a finitely-supported one. And here’s the graph: its nodes are pairs of distinct atoms, and there is an edge from to , whenever are distinct atoms. That’s it.