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 X and Y be two orbit-finite equivariant sets with atoms. If there exists a finitely supported bijection f \colon X \rightarrow Y then there exists an equivariant bijection F \colon X \rightarrow Y.
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Join of two elements

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

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

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Bipartite graphs without bipartite partition

In the equality symmetry, there is a single-orbit graph G=(V,E) with the property that G 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 G 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 (a,b) to (b,a), whenever a,b are distinct atoms. That’s it.

Color-preserving automorphisms in Fraisse classes (new version)

Again, a question closely related to a characterization of standard atoms (not to be confused with standard alphabets:). The question is a refinement of a question posted in Automorphisms vs color-preserving automorphisms in Fraisse classes, which has been answered negatively (see the recent post The colored open question closed).

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