arXiv subjects

Rehana Patel

Publications and source records attributed to Rehana Patel.

14 recordsLinked to original sources

arXiv2026

Forcing with Invariant Measures

This paper introduces a model-theoretic generalization of the notion of forcing with random reals, in which forcing gives rise to random generic structures. Specifically, we consider forcing with $\kappa$-Borel probability measures on the space of $L$-structures with a (possibly uncountable) infinite set $X$, focusing on those that are invariant under the action of the symmetric group $Sym(X)$. We demonstrate how any $Sym(X)$-invariant measure where $X$ is countable can be uniquely extended to a $Sym(Y)$-invariant measure where $Y$ is uncountable, and prove that forcing with such measures satisfies the countable chain condition. We also show that we can uniformly distinguish between these random generic structures and the Cohen generic structures that arise from forcing with a strong Fra\"iss\'e class: There is a $\kappa$-Borel set of low complexity that contains every Cohen generic structure that is not highly homogeneous but contains no random generic structure, implying that a structure that is not highly homogeneous cannot be both Cohen generic and random generic. Finally, we answer an open question of Kostana in the case of $\omega_1$, by establishing a connection between forcing with a strong Fra\"iss\'e class and Cohen forcing.

math.LO↗

arXiv2025

Big Ramsey Degrees in Ultraproducts of Finite Structures

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $\fLL^*$ has, as a spine, $η_1$, an uncountable analogue of the order type of rationals $η$. Finite big Ramsey degrees for $η$ were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that $η_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to $η_1$ to show that it witnesses big Ramsey degrees of finite tuples in $η$ on every copy of $η$ in $η_1,$ and consequently in $\fLL^*$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

math.LO↗

arXiv2025

Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

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.

math.CO↗

arXiv2022

Fraisse Structures with SDAP+, Part I: Indivisibility

This is Part I of a two-part series regarding Ramsey properties of Fraisse structures satisfying a property called SDAP+, which strengthens the Disjoint Amalgamation Property. We prove that every Fraisse structure in a finite relational language with relation symbols of any finite arity satisfying this property is indivisible. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct proof of indivisibility which uses the method of forcing to conduct unbounded searches for finite sets. In Part II, we prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Part II utilizes a theorem from Part I as a pigeonhole principle for induction arguments. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures.

math.CO↗

arXiv2022

Fraisse structures with SDAP+, Part II: Simply characterized big Ramsey structures

This is Part II of a two-part series regarding Ramsey properties of Fraisse structures satisfying a property called SDAP+, which strengthens the Disjoint Amalgamation Property. In Part I, we prove that every Fraisse structure in a finite relational language with relation symbols of any finite arity satisfying this property is indivisible. In Part II, we prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Part II utilizes the notion of coding trees of 1-types developed in Part I and a theorem from Part I which functions as a pigeonhole principle for induction arguments in this paper. Our approach yields a direct characterization of the degrees without appeal to the standard method of "envelopes". This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures.

math.CO↗

arXiv2021

Fraisse classes with simply characterized big Ramsey degrees

We formulate a property strengthening the Disjoint Amalgamation Property and prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Furthermore, we prove indivisibility for every Fraisse structure in an arbitrary finite relational language satisfying this property. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct characterization of the degrees without appeal to the standard method of "envelopes".

math.CO↗

arXiv2021

On computable aspects of algebraic and definable closure

We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $Σ^0_{n+2}$ sets. We further show that these bounds are tight.

math.LO↗

arXiv2019

The entropy function of an invariant measure

Given a countable relational language $L$, we consider probability measures on the space of $L$-structures with underlying set $\mathbb{N}$ that are invariant under the logic action. We study the growth rate of the entropy function of such a measure, defined to be the function sending $n \in \mathbb{N}$ to the entropy of the measure induced by restrictions to $L$-structures on $\{0, \ldots, n-1\}$. When $L$ has finitely many relation symbols, all of arity $k\ge 1$, and the measure has a property called non-redundance, we show that the entropy function is of the form $Cn^k+o(n^k)$, generalizing a result of Aldous and Janson. When $k\ge 2$, we show that there are invariant measures whose entropy functions grow arbitrarily fast in $o(n^k)$, extending a result of Hatami-Norine. For possibly infinite languages $L$, we give an explicit upper bound on the entropy functions of non-redundant invariant measures in terms of the number of relation symbols in $L$ of each arity; this implies that finite-valued entropy functions can grow arbitrarily fast.

math.LO↗

arXiv2018

Stable regularity for relational structures

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a structure has an equitable regularity partition of size polynomial in the reciprocal of the desired accuracy, and such that for each $k$-ary relation and $k$-tuple of parts of the partition, the density is close to either 0 or 1. In addition, we provide regularity results for finite and Borel structures that satisfy a weaker notion that we call almost stability.

math.LO↗

arXiv2017

Properly ergodic structures

We consider ergodic $\mathrm{Sym}(\mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $\mathbb{N}$ (for $L$ a countable relational language), and call such a measure a properly ergodic structure when no isomorphism class of structures is assigned measure $1$. We characterize those theories in countable fragments of $\mathcal{L}_{ω_1, ω}$ for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment $F$ of $\mathcal{L}_{ω_1, ω}$ the almost-sure $F$-theory of a properly ergodic structure has continuum-many models (an analogue of Vaught's Conjecture in this context), but its full almost-sure $\mathcal{L}_{ω_1, ω}$-theory has no models. We also show that, for an $F$-theory $T$, if there is some properly ergodic structure that concentrates on the class of models of $T$, then there are continuum-many such properly ergodic structures.

math.LO↗

arXiv2017

Countable infinitary theories admitting an invariant measure

Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $Σ$ of $\mathcal{L}_{ω_1, ω}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection of models of $Σ$ with underlying set $\mathbb{N}$. Restricting to $\mathcal{L}_{ω, ω}(L)$, this answers an open question of Gaifman from 1964, via a translation between $S_\infty$-invariant measures and Gaifman's symmetric measure-models with strict equality. It also extends the known characterization in the case where $Σ$ implies a Scott sentence. To establish our result, we introduce machinery for building invariant measures from a directed system of countable structures with measures.

math.LO↗

arXiv2016

A classification of orbits admitting a unique invariant measure

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_\infty$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\mathbb{Q}, <)$.

math.LO↗

arXiv2016

Invariant measures concentrated on countable structures

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraisse limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

math.LO↗

arXiv2015

Invariant measures via inverse limits of finite structures

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite structures. We use this construction to obtain invariant probability measures concentrated on the classes of countable models of certain first-order theories, including measures that do not assign positive measure to the isomorphism class of any single model. We also characterize those transitive Borel G-spaces admitting a G-invariant probability measure, when G is an arbitrary countable product of symmetric groups on a countable set.

math.LO↗