When invariance implies exchangeability (and applications to invariant Keisler measures)

TL;DR

The paper investigates when Aut$(rak M)$‑invariant measures on expansions of a countable homogeneous structure must be exchangeable under all permutations of the domain. It develops a combinatorial criterion based on $k$‑overlap closed ages and slow growth conditions that, for many base structures (e.g., finitely bounded homogeneous 3‑hypergraphs with free amalgamation), forces exchangeability and hence enables explicit Aldous–Hoover representations. It then applies these results to invariant Keisler measures, showing many such measures arise from invariant random expansions and describing their spaces in several important homogeneous structures; it also exposes natural exceptions via parity hypergraphs where non‑exchangeable invariant random expansions exist. The work provides, for the first time in some high‑arity homogeneous contexts, a coherent framework linking exchangeability, Fraïssé theory, and invariant Keisler measures, with broader implications for model theory and probabilistic combinatorics. It further identifies phenomena (e.g., universal measure zero formulas versus forking) that reveal rich and nuanced behaviour of invariant measures outside NIP, offering several directions for future research in higher arity and more general automorphism groups.

Abstract

We study the problem of when, given a countable homogeneous structure $M$ and a space $S$ of expansions of $M$, every $\mathrm{Aut}(M)$-invariant probability measure on $S$ is exchangeable (i.e. invariant under all permutations of the domain). We show, for example, that if $M$ is a finitely bounded homogeneous $3$-hypergraph with free amalgamation (including the generic tetrahedron-free $3$-hypergraph), all $\mathrm{Aut}(M)$-invariant random expansions by graphs are exchangeable. Moreover, we extend and recover both the work of Angel, Kechris, and Lyons on invariant random orderings and some of the work of Crane and Towsner, and Ackerman on relative exchangeability. In the second part of the paper, we apply our results to the study of invariant Keisler measures, which we prove to be particular invariant random expansions. Thus, we describe the spaces of invariant Keisler measures of various homogeneous structures, obtaining the first results of this kind since the work of Albert and Ensley. We also show there are $2^{\aleph_0}$ supersimple homogeneous ternary structures for which there are non-forking formulas which are universally measure zero.

Figures (3)

• Figure 3: A pictorial representation of $k$-overlap closedness of $\mathcal{C}$: for some dense enough $r$-hypergraph $\mathbb{K}$ on $n$ vertices where no two hyperedges intersect in more than $k$ vertices, for any two $H_1, H_2\in\mathcal{C}[r]$, any way of pasting $H_1$ and $H_2$ into the $r$-hyperedges of $\mathbb{K}$ gives some $G\in\mathcal{C}[n]$ (after possibly adding some relations that do not change the induced structures on the $r$-hyperedges of $\mathbb{K}$).
• Figure 4: The two starting configurations for the case of a $3$-uniform hypergraph with $r=3$. In Case 1, $b_3$ needs to be connected to each of $\{a_2, a_3\}$, and by Lemma \ref{['lemma:Steiner']} (3) each such hyperedge must contain a new point. In Case 2, $b_3$ needs to be connected to each of $\{a_2, a_3\}$, while $b'_3$ needs to be connected to each of $\{a_1, a_2\}$, and at least three such hyperedges must each contain a new point. Either case leads to indefinitely creating new hyperedges with new points until the size of the forbidden structures is exceeded.
• Figure 5: Simplified map of the model-theoretic universe. Note that $\mathrm{stable}\subsetneq\mathrm{simple}\subsetneq\mathrm{NSOP}$ and $\mathrm{NIP}\cap\mathrm{NSOP}=\mathrm{stable}$. The map includes examples of homogeneous (or finitely homogenisable) structures in the various classes. We already introduced the strictly simple and strictly $\mathrm{NSOP}$ ones in Example \ref{['ex:hypergraphs']}. In the strictly $\mathrm{NIP}$ part of the universe we can see the dense linear order $(\mathbb{Q},<)$, the cyclical order $(\mathbb{Q}, \mathrm{cyc})$, and Droste's $2$-homogeneous semilinear orders droste1985structure, denoted by $(\mathcal{T}, \leq)$. In the stable part of the universe, we have an infinite set with equality and the countable disjoint union of copies of the complete graph $K_n$. An example of a homogeneous structure outside all of these classes is universal homogeneous ordered $r$-hypergraph $(\mathcal{R}_r, <)$.

Theorems & Definitions (135)

• Theorem A: Theorem \ref{['thm:cre si']}
• Theorem B: Theorem \ref{['thm:appliedforIKM']}
• Theorem C: Theorem \ref{['IKMdichotomy']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.6: disjoint amalgamation and free amalgamation
• Definition 2.7
• Definition 2.8