Generic sampling and invariant measures on the space of $k$-uniform hypergraphs
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917027038363648 |
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| author | Ackerman, Nathanael Freer, Cameron Gannon, Kyle Hanson, James E. Patel, Rehana |
| author_facet | Ackerman, Nathanael Freer, Cameron Gannon, Kyle Hanson, James E. Patel, Rehana |
| contents | We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure $μ_{W}$ such that generic sampling with respect to $μ_{W}$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k = 2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generic sampling and invariant measures on the space of $k$-uniform hypergraphs Ackerman, Nathanael Freer, Cameron Gannon, Kyle Hanson, James E. Patel, Rehana Combinatorics Logic Probability We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure $μ_{W}$ such that generic sampling with respect to $μ_{W}$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k = 2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph. |
| title | Generic sampling and invariant measures on the space of $k$-uniform hypergraphs |
| topic | Combinatorics Logic Probability |
| url | https://arxiv.org/abs/2510.17090 |