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Sergei Starchenko

Publications and source records attributed to Sergei Starchenko.

At least 19 recordsLinked to original sources
arXiv2026

Sum-product Phenomenon Via Dimension

We show a sum-product phenomenon in fields equipped with abstract dimension theories, which simultaneously generalizes the dimensions in geometric theories and Hrushovski's coarse pseudo-finite dimensions. More precisely, we show that for type-definable sets of positive non-zero dimension, non-expansion in dimension of both the sumset and product set implies the existence of a definable field in the same dimension. Main ingredients of the proof include dimensional analogues of the Ruzsa triangle inequality and the Pl\"unnecke-Ruzsa inequality.

math.LO↗
arXiv2026

Combinatorics in one-based and related structures

We consider some extremal combinatorial questions for bipartite graphs definable in stable one-based (and related) structures. We show that they satisfy both strong Erd\H{o}s-Hajnal property and linear Zarankiewicz. We also show that the same is true for both collapsed and uncollapsed Hrushovski's ``ab initio'' constructions, and discuss some connections to Zilber's trichotomy principle. For strong Erd\H{o}s-Hajnal, we show that in fact it holds in a more general class of $1$-semi-equational theories.

math.LO↗
arXiv2024

Limits of definable families and dilations in nilmanifolds

Let $G$ be a unipotent group and $\mathcal F=\{F_t:t\in (0,\infty)\}$ a family of subsets of $G$, with $\mathcal F$ definable in an o-minimal expansion of the real field. Given a lattice $\Gamma\subseteq G$, we study the possible Hausdorff limits of $\pi(\mathcal F)$ in $G/\Gamma$ as $t$ tends to $\infty$ (here $\pi:G\to G/\Gamma$ is the canonical projection). Towards a solution, we associate to $\mathcal F$ finitely many real algebraic subgroups $L\subseteq G$, and, uniformly in $\Gamma$, determine if the only Hausdorff limit at $\infty$ is $G/\Gamma$, depending on whether $L^\Gamma=G$ or not. The special case of polynomial dilations of a definable set is treated in details.

math.LO↗
arXiv2021

Zarankiewicz's problem for semilinear hypergraphs

A bipartite graph $H = \left(V_1, V_2; E \right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i \subseteq \mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of $s$ linear equalities and inequalities in $d_1 + d_2$ variables for some $s$. We show that for a fixed $k$, the number of edges in a $K_{k,k}$-free semilinear $H$ is almost linear in $n$, namely $|E| = O_{s,k,\varepsilon}(n^{1+\varepsilon})$ for any $\varepsilon > 0$; and more generally, $|E| = O_{s,k,r,\varepsilon}(n^{r-1 + \varepsilon})$ for a $K_{k, \ldots,k}$-free semilinear $r$-partite $r$-uniform hypergraph. As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis parallel sides in $\mathbb{R}^d$ such that their incidence graph is $K_{k,k}$-free, there can be at most $O_{k,\varepsilon}(n^{1+\varepsilon})$ incidences. The same bound holds if instead of boxes one takes polytopes cut out by the translates of an arbitrary fixed finite set of halfspaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in $o$-minimal structures (showing that the failure of an almost linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

math.CO↗
arXiv2021

Peterzil-Steinhorn subgroups and $μ$-stabilizers in ACF

We consider $G$, a linear group defined over $k$, an algebraically closed field. By considering $k$ as an embedded residue field of an algebraically closed valued field $K$, we can associate to it a compact $G$-space $S^μ_G(k)$, consisting of $μ$-types on $G$. We showed that for each $p_μ\in S^μ_G(k)$, $\text{Stab}^μ(p)=\text{Stab}(p_μ)$ is a solvable infinite algebraic group when $p_μ$ is centered at infinity and residually algebraic. Moreover we give a description of the dimension $\text{Stab}(p_μ)$ in terms of dimension of $p$.

math.LO↗
arXiv2021

O-minimal flows on nilmanifolds

Let $G$ be a connected, simply connected nilpotent Lie group, identified with a real algebraic subgroup of $\mathrm{UT}(n,\mathbb{R})$, and let $Γ$ be a lattice in $G$, with $π:G\to G/Γ$ the quotient map. For a semi-algebraic $X\subseteq G$, and more generally a definable set in an o-minimal structure on the real field, we consider the topological closure of $π(X)$ in the compact nilmanifold $G/Γ$. Our theorem describes $\mathrm{cl}(π(X))$ in terms of finitely many families of cosets of real algebraic subgroups of $G$. The underlying families are extracted from $X$, independently of $Γ$. We also prove an equidistribution result in the case of curves.

math.LO↗
arXiv2021

Model-theoretic Elekes-Szab\'o for stable and o-minimal hypergraphs

A theorem of Elekes and Szab\'{o} recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in: 1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic $0$); and 2) $o$-minimal expansions of groups. Our methods provide explicit bounds on the power saving exponent in the non-group case. Ingredients of the proof include: a higher arity generalization of the abelian group configuration theorem in stable structures, along with a purely combinatorial variant characterizing Latin hypercubes that arise from abelian groups; and Zarankiewicz-style bounds for hypergraphs definable in distal structures.

math.LO↗
arXiv2021

Definable regularity lemmas for NIP hypergraphs

We present a systematic study of the regularity phenomena for NIP hypergraphs and connections to the theory of (locally) generically stable measures, providing a model-theoretic hypergraph version of the results from [L. Lovász, B. Szegedy, "Regularity partitions and the topology of graphons", An irregular mind, Springer Berlin Heidelberg, 2010, 415-446]. Besides, we revise the two extremal cases of regularity for stable and distal hypergraphs, improving and generalizing the results from [A. Chernikov, S. Starchenko, "Regularity lemma for distal structures", J. Eur. Math. Soc. 20 (2018), 2437-2466] and [M. Malliaris, S. Shelah, "Regularity lemmas for stable graphs", Transactions of the American Mathematical Society, 366.3, 2014, 1551-1585]. Finally, we consider a related question of the existence of large (approximately) homogeneous definable subsets of NIP hypergraphs and provide some positive results and counterexamples.

math.LO↗
arXiv2020

Cutting lemma and Zarankiewicz's problem in distal structures

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we generalize the results in [J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl. "A semi-algebraic version of Zarankiewicz's problem"] on the semialgebraic planar Zarankiewicz problem to arbitrary $o$-minimal structures, in particular obtaining an $o$-minimal generalization of the Szemerédi-Trotter theorem.

math.LO↗
arXiv2018

Ramsey growth in some NIP structures

We investigate bounds in Ramsey's theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek [B. Bukh, J. Matoušek. "Erdős-Szekeres-type statements: Ramsey function and decidability in dimension $1$", Duke Mathematical Journal 163.12 (2014): 2243-2270] from the semialgebraic case to arbitrary polynomially bounded $o$-minimal expansions of $\mathbb{R}$, and show that it doesn't hold in $\mathbb{R}_{\exp}$. This provides a new combinatorial characterization of polynomial boundedness for $o$-minimal structures. We also prove an analog for relations definable in $P$-minimal structures, in particular for the field of the $p$-adics. Generalizing [D. Conlon, J. Fox, J. Pach, B. Sudakov, A. Suk "Ramsey-type results for semi-algebraic relations", Transactions of the American Mathematical Society 366.9 (2014): 5043-5065], we show that in distal structures the upper bound for $k$-ary definable relations is given by the exponential tower of height $k-1$.

math.LO↗
arXiv2017

Algebraic and o-minimal flows on complex and real tori

We consider the covering map $π:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $π(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.

math.AG↗
arXiv2016

A note on the Erdős-Hajnal property for stable graphs

In this short note we provide a relatively simple proof of the Erdős-Hajnal conjecture for families of finite (hyper-)graphs without the $k$-order property. It was originally proved by M. Malliaris and S. Shelah in "Regularity lemmas for stable graphs", Transactions AMS, 366, 2014, 1551-1585.

math.LO↗
arXiv2016

Regularity lemma for distal structures

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into very homogeneous semialgebraic pieces up to a small error (e.g., see [33, 2, 16, 18]). We show that similar results can be obtained for families of graphs with the edge relation uniformly definable in a structure satisfying a certain model theoretic property called distality, with respect to a large class of generically stable measures. Moreover, distality characterizes these strong regularity properties. This applies in particular to graphs definable in arbitrary $o$-minimal structures and in $p$-adics.

math.LO↗
arXiv2015

Topological groups, μ-types and their stabilizers

We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of partial types $S^μ_G(M)=\{p_μ:p\in S_G(M)\}$ which is the quotient of the usual type space $S_G(M)$ by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if $p$ is a definable type then it has a corresponding definable subgroup $Stab_μ(p)$, which is the stabilizer of $p_μ$. This group is nontrivial when $p$ is unbounded in the sense of $\mathcal M$; in fact it is a torsion-free solvable group. Along the way, we analyze the general construction of $S^μ_G(M)$ and its connection to the Samuel compactification of topological groups.

math.LO↗