arXiv subjects

Mostafa Mirabi

Publications and source records attributed to Mostafa Mirabi.

At least 19 recordsLinked to original sources

arXiv2026

Almost-linear Zarankiewicz bounds in $1$-semi-equational theories

We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity $r\geq2$. If $T$ is a $1$-semi-equational theory, then, for every formula $φ$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-free $r$-partite hypergraph defined by $φ$ on $n$ vertices has $O_{T,φ,t,r}\!\left( n^{r-1}(1+\log(1+n))^c \right) $ edges. Put $α_k=\min\{k-1,2\}$. In the bipartite case, a Boolean combination of $m$ $(k,1)$-semi-equations has $ O_{k,t,m}\!\left( n(1+\log(1+n))^{(m-1)α_k} \right) $ edges whenever it is $K_{t,t}$-free. In particular, a relation defined by one $(k,1)$-semi-equation or its negation has a linear bound. The proofs combine incidence estimates for indexed set systems with low-crossing orderings of finite $k$-wise laminar families. Consequently, no $1$-semi-equational theory locally trace-defines an infinite domain.

math.LO↗

arXiv2026

Relative stability, relative categoricity and internal covers

Let $T$ be a countable complete theory with a distinguished unary predicate $P$, and let $T^{P}$ be the theory of the $P$-parts of models of $T$ with the induced structure. $T$ is said to be relatively categorical or categorical over $P$ if any isomorphism between the $P$-parts of two models of $T$ lifts to an isomorphism of the models in question. We study the special case of relative categoricity where $T$ is internal to $T^{P}$ (that is, every model $M$ of $T$ is in the definable closure of $P(M)$ together with additional parameters from $M$). We first give a structure theory for such $T$: after passing to $T^{eq}$ and naming a parameter, $T$ is the same thing as a "pure torsor cover" of $T^{P}$, namely simply adjoining to $T^{P}$ a new sort for a torsor $S$ for a $\emptyset$-definable group $G$ in $T^{P}$, with no additional structure. We discuss relative stability, or stability over $P$, and give a characterization of relative stability, superstability, and $ω$-stability of $T$ in terms of $H$ having the stable chain condition, superstable chain condition, and $ω$-stable chain condition, respectively.

math.LO↗

arXiv2026

One-point extensions of Euclidean Ramsey sets

Let $X$ be a finite Euclidean Ramsey set. We prove that adjoining any point outside the affine hull of $X$ gives another Euclidean Ramsey set, answering a conjecture of Ivan, Leader, and Walters. We first give an elementary product proof under the additional assumptions that the orthogonal projection of the new point lies in $conv(X)$ and that its distance from $aff(X)$ is sufficiently large. We then prove the general case by combining the product theorem for $E$-Ramsey configurations and Kř\'ıž's orbit-gluing theorem with a cyclic construction. No transitivity assumption on $X$ is needed.

math.CO↗

arXiv2026

First-Order Laws for Random Geometric Graphs on the Torus

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.

math.PR↗

arXiv2026

Canonical Expansions of $\mathbb R_{\mathcal Q}$-Germs

Let $\mathcal{H}_{\mathcal{Q}}$ be the field of unary germs at $0^+$ definable in the quasianalytic o-minimal structure $\mathbb{R}_{\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of $\mathcal{H}_{\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that $\mathbb{R}_{\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic $\mathbb{R}_{\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary $\mathbb{R}_{\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of $\mathcal{H}_{\mathcal{Q}}$.

math.LO↗

arXiv2026

Asymptotic classes via definable quotients

We study when finite-dimensional asymptoticity transfers through uniform interpretations and definable quotients. Ambient counting yields finitely many asymptotic alternatives, but the corresponding parameter cells may fail to be definable in the interpreted language; in general, one obtains only weak asymptoticity. We isolate "trace reflection", a one-sided descent condition that removes this obstruction. For interpretations with definable selectors, trace reflection transfers asymptoticity with explicit denominator bounds. For general definable quotients, invariant asymptotic profiles replace definable choice and yield a quotient-transfer theorem. We identify an intrinsic visible denominator and prove that it is the least possible asymptotic denominator, with a coprime-witness criterion for when the interpretation-dependent bound is sharp. We also establish formula-wise, syntactic, and semantic descent criteria, prove composition theorems, and show that trace reflection is strictly weaker than uniform weak bi-interpretability. A framed-profile argument removes the noncanonical coordinate parameters introduced by uniform finite-field reconstruction. As applications, we study projective quotient traces and prove that, for every fixed $k\geq2$, the pure incidence structures arising from graphs of polynomials of degree less than $k$ over finite fields form a full $k$-dimensional asymptotic class. The denominator $k$ is minimal, and every member is $K_{k,n}$-free for all $n\geq2$.

math.LO↗

arXiv2026

MS-measurability via Coordinatization

We prove that Macpherson-Steinhorn measurability is preserved under finite nil-interaction tree products. If the canonical component theories are MS-measurable and their universes are normalized to have dimension one, then every admissible nil-interaction coordinatized structure is MS-measurable. The resulting dimension is the sum of the local component dimensions appearing in the tree closure, and the measure is the corresponding product of local measures. We also prove a structural converse: in the countable $\aleph_0$-categorical setting, admissible nil-interaction structures are precisely definable expansions of their canonical tree products. Thus nil-interaction exactly isolates the tree-product case of coordinatization. Further preservation results show that canonical tree products preserve supersimplicity of finite SU-rank and one-basedness, with SU-rank computed by the same local sum formula. Finite homogeneous component approximations also combine into finite tree envelopes, so smooth approximability is preserved. Finally, enriched finite tree products of multidimensional asymptotic, respectively exact, component classes are again multidimensional asymptotic, respectively exact, up to the natural weak/reduct distinction.

math.LO↗

arXiv2026

Limit laws for component-pruned sparse random graphs and percolated tori

We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erdős-Rényi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfies a zero-one law for $\mathrm{MSO}_2$, with quantification over sets of vertices and sets of edges. The proof combines uniform component counts, an MSO Feferman-Vaught decomposition for disjoint unions, and semilinearity of the order spectra of MSO-definable classes of finite trees. We also show that the term $f(n)\log f(n)$ cannot simply be omitted: star components can occur at first-order-visible Poisson thresholds. We further establish first-order limit laws for bond percolation on the discrete torus $T_L^d$. In the two-sided subpolynomial regime, pruning below a sufficiently slow threshold yields a zero-one law. For the unpruned model in either one-sided polynomial regime, the reciprocal exponents $α=1/k$ are precisely the critical scales. At such a scale, an extended limit of $N p_N^k$ or $N q_N^k$ equal to $0$ or $\infty$ gives a zero-one law; a positive finite limit gives a convergence law but not a zero-one law; and the absence of an extended limit gives failure of convergence. Finally, $\mathrm{MSO}_1$ already detects the parity of the torus side length through bipartiteness, producing a natural obstruction to monadic convergence in a near-deterministic regime.

math.LO↗

arXiv2026

From Shelah's block-content to Hales-Jewett

We study the quantitative relationship between the Hales-Jewett numbers and Shelah's block-content canonization functions. Block-content canonization yields a block subspace on which the color of a word is determined solely by the multiplicities of the alphabet letters among the variable blocks. We show that this canonical information, combined with the multidimensional Gallai-Witt theorem, suffices to produce a monochromatic Hales-Jewett subspace. The argument passes to the space of content vectors, finds a monochromatic homothetic copy of a finite content simplex, and lifts it through the canonical block subspace. Combined with the elementary fact that Hales-Jewett bounds block canonization, this gives a two-way quantitative comparison up to an explicit change of parameters. The underlying mechanism may be summarized by the slogan Hales-Jewett = block canonization + Gallai-Witt We also prove the corresponding equal-block result and show, by an explicit coloring over every finite field of odd prime order, that the analogous singleton-coordinate canonization principle fails as soon as two coordinates remain live.

math.CO↗

arXiv2026

Scott complexity of trees of finite rank via degrees of categoricity

In earlier work \cite{Mah19}, the first author constructed, for each finite $m\geq 1$, a computable tree $\A_{m+1}$ of rank $m+1$ whose strong degree of categoricity is $\mathbf{0}^{(2m)}$, and showed this degree is optimal at each rank. Those results are lightface: they concern Turing degrees of isomorphisms between computable copies. In this paper we determine the boldface content of the construction. We isolate a transfer principle: a degree-of-categoricity lower bound that holds uniformly relative to every oracle defeats $L_{ω_1ω}$-definability of automorphism orbits outright. We verify that the construction of \cite{Mah19} has this uniformity, and deduce that $\A_{m+1}$ has Scott rank exactly $2m+1$, with Scott sentence complexity one of $\Sin{2m+1}$, $\dSin{2m+1}$, or $\Pin{2m+2}$. For rank $2$ we carry out a complete, computability-free Scott analysis: the orbit of the level-one nodes of infinite degree is $\Pin{2}$- but not $\Sin{2}$-definable, and $\SSC(\A_2)=\Pin{4}$ exactly, witnessed by a computable $\Pc{4}$ Scott sentence. We then prove $\SSC(\A_{m+1})=\Pin{2m+2}$ for all finite $m$: among the three candidates, only $\Pin{2m+2}$ is consistent with the parameterized Scott rank, and a parameter-reservation argument -- naming any finite tuple reserves only finitely many top-level subtrees, and the relativized coding survives on the infinitely many spare subtrees -- pins $\pSR(\A_{m+1})=2m+1$, selecting that candidate. These results begin a classification of the Scott sentence complexities of trees of finite rank, in analogy with the Gonzalez--Rossegger analysis of linear orders, and connect the $2α$-jump phenomenon in degrees of categoricity to Scott spectral-gap questions for trees.

math.LO↗

arXiv2026

The Failure of Stable Composition for Equivalence Relations in Simple Theories

Casanovas and Potier proved that algebraic quantification preserves stability of formulas. They also gave a nonsimple example, answering a question of Laskowski, showing that the algebraicity hypothesis cannot simply be replaced by NFCP, and asked whether a similar example exists in a simple theory. We give such an example in elementary form. The edge-set structure of the random bipartite graph has two definable equivalence relations, both stable and NFCP, whose relational composition has the order property. The resulting theory is simple and $\aleph_0$-categorical. We also prove a formal sharpness observation: every formula in every first-order theory is an existential composition of two stable NFCP formulas algebraic in the two outer variables. Consequently, closure of stable NFCP formulas under this composition characterizes stability of the whole theory.

math.LO↗

arXiv2026

Level Totients for Integer Mosaics

We study a level analog of Euler's totient function for integer mosaics. Let $P_i(n)$ be the set of primes appearing in the first $i$ levels of the mosaic of $n$, and let $φ_i(n)$ count the integers $m\leq n$ for which $P_i(m)\cap P_i(n)=\varnothing$. We prove a Möbius divisor-sum formula for $φ_i(n)$ and reduce it to a sum over a set $\mathcal{V}_{i,S}$ of powerful integers. If $S\neq\varnothing$, $i\geq 2$, and $q=\min S$, then \[ |\mathcal{V}_{i,S}\cap[1,x]|\sim C_{i,S}x^{1/q}, \] with $C_{i,S}$ an explicit positive Euler-product constant. For fixed $S$, the density $δ_i(S)$ of integers whose first $i$ levels avoid $S$ exists and has an Euler product; for nonempty $S$, $i\ge2$, and $q=\min S$, the number of such integers up to $N$ is $δ_i(S)N+O_{i,S}(N^{1/q})$. Taking $S=P_i(n)$ gives \[ \frac{φ_i(n)}{n}=δ_i(P_i(n))+O_\varepsilon(n^{-1/2+\varepsilon}) \] uniformly in $n$.

math.NT↗

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 $κ$-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 Fraïssé class: There is a $κ$-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 $ω_1$, by establishing a connection between forcing with a strong Fraïssé class and Cohen forcing.

math.LO↗

arXiv2026

On the computability of cofinal Fraïssé limits

For any collection of finite structures closed under isomorphism (i.e., an age) which has the Hereditary Property (HP), the Joint Embedding Property (JEP), and the Cofinal Amalgamation Property (CAP), there is a unique (up to isomorphism) countable structure which is cofinally ultrahomogeneous with the given age. Such a structure is called the cofinal Fraïssé limit of the age. In this paper, we consider the computational strength needed to construct the cofinal Fraïssé limit of a computable age. We show that this construction can always be done using the oracle 0''', and that there are ages that require 0''. In contrast, we show that if one assumes the strengthening of (CAP) known as the Amalgamation Property (AP), then the resulting limit, called the Fraïssé limit, can be constructed from the age using 0'. Our results therefore show that the more general case of cofinal Fraïssé limits requires greater computational strength than Fraïssé limits.

math.LO↗

arXiv2025

Structured Sunflowers

We call an infinite structure $\mathcal{M}$ sunflowerable if whenever $\mathcal{M}'$ is isomorphic to $\mathcal{M}$ with underlying set $M'$, consisting of finite sets of bounded size, there is an $M_0 \subseteq M'$ such that $M_0$ is a sunflower and $\mathcal{M}'\!\!\upharpoonright[M_0]$ is isomorphic to $\mathcal{M}$. We give sufficient conditions on $\mathcal{M}$ to show that $\mathcal{M}$ is sunflowerable. These conditions allow us to show that several well-known structures are sunflowerable and give a complete characterization of the countable linear orderings which are sunflowerable. We show that a sunflowerable structure must be indivisible. This allows us to show that any Fraïssé limit which has the 3-disjoint amalgamation property and a single unary type must be indivisible. In addition to studying sunflowerability of infinite structures, we also consider an analogous property of an age which we call the sunflower property. We show that any sunflowerable structure must have an age with the sunflower property. We also give concrete bounds in the case that the age has the hereditary property, the 3-disjoint amalgamation property, and is indivisible.

math.CO↗

arXiv2025

Rainbow Threshold Graphs

We define a generalization of threshold graphs which we call $k$-rainbow threshold graphs. We show that the collection of $k$-rainbow threshold graphs do not satisfy the $0$-$1$ law for first order logic and that asymptotically almost surely all $(k+1)$-rainbow threshold graphs are not isomorphic to a $k$-rainbow threshold graph.

math.CO↗

arXiv2025

Absoluteness of Fixed Points

We characterize those complete commutative positive linear ordered monoids $W$ such that whenever $f$ is a map from a Cauchy complete $W$-metric space to itself, the existence of a fixed point of $f$ is independent of the background model of set theory.

math.GN↗

arXiv2024

Cohen Generic Structures with Functions

Suppose $\mathscr{L}^-\subseteq \mathscr{L}$ are languages where $\mathscr{L} \setminus\mathscr{L}^-$ is relational. Additionally, let $\mathbf{K}$ be a strong $\textrm{Fraïssé}$ class in $\mathscr{L}$. We consider the partial ordering, under substructure, of those elements in $\mathbf{K}$ whose reduct to $\mathscr{L}^-$ are substructures of a fixed $\mathscr{L}^-$-structure $\mathcal{M}^-$. In this paper, we establish that, under general conditions, this partial order satisfies the $|\mathcal{M}^-|$-chain condition. Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse\ limit of $\mathbf{K}$, provided $\mathcal{M}^-$ satisfies the theory of$\textrm{Fraïssé}$ limit of its age. We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups.

math.LO↗