Doug Ensley, Rami Grossberg
We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.
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We examine a version of Ramsey's theorem based on Tao, Gaspar and Kohlenbach's "finitary" infinite pigeonhole principle.We will show that the "finitary" infinite Ramsey's theorem naturally gives rise to statements at the level of the infinite Ramsey's theorem, Friedman's infinite adjacent Ramsey theorem (well-foundedness of certain ordinals up to $\varepsilon_0$), $1$-consistency of theories up to PA and the finite Ramsey's theorem.
In a recent paper \cite{So} S. Solecki proves a finite self dual Ramsey theorem that in a natural way gives simultaneously the classical finite Ramsey theorem \cite{Ra} and the Graham-Rothschild theorem \cite{Gr-Ro}. In this paper we prove the corresponding infinite dimensional self dual theorem, giving similarly as a consequence the infinite classical Ramsey theorem \cite{Ra} and the Carlson-Simpson theorem \cite{Ca-Si}.
We show that, assuming GCH, if $κ$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $κ$ and obeying the constraints of Easton's theorem, namely, $F(α)\leq F(β)$ for $α\leqβ$ and $α<\cf(F(α))$, then there is a cofinality preserving forcing extension in which $κ$ remains Ramsey or strongly Ramsey respectively and $2^δ=F(δ)$ for every regular cardinal $δ$.
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable reducibility. We proceed to a systematic study of various Ramsey-type statements under a unified and minimalistic framework and obtain a precise analysis of their interrelations. We clarify the role of the number of colors in Ramsey's theorem. In particular, we show that the hierarchy of Ramsey's theorem induced by the number of colors is strictly increasing over computable reducibility, and exhibit …
We work within the framework of the Alpha-Theory introduced by Benci and Di Nasso. The Alpha-Theory postulates a few natural properties for an infinite "ideal" number $α$. The formulation provides an elementary axiomatics for the methods of abstract ultra-Ramsey theory. The main results are Theorem 10, Theorem 57, Theorem 67 and Theorem 73. Theorem 10 is an infinite-dimensional extension of the celebrated Ramsey's Theorem. We show that corollaries of this result include the Galvin-Pirky Theorem, the Silver Theorem and the $\vecα$-Ellentuck Theorem. We prove that, under the assumption of the $\…
We study the positions in the Weihrauch lattice of parallel products of various combinatorial principles related to Ramsey's theorem. Among other results, we obtain an answer to a question of Brattka, by showing that Ramsey's theorem for pairs ($\mathsf{RT}^2_2$) is strictly Weihrauch below the parallel product of the stable Ramsey's theorem for pairs and the cohesive principle ($\mathsf{SRT}^2_2 \times \mathsf{COH}$).
We give a characterizations of Ramsey ultrafilters on $\mathscr P(ω)$ in terms of functions $f:ω^n\toω$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[ω]^n$ there is a finite partition~$\mathcal Q$ of $[ω]^{2n}$ such that any set $X\subseteq ω$ that is homogeneous for $\mathcal Q$ is a finite union of sets that are canonical for $\mathcal P$.
We study the uniform computational content of Ramsey's theorem in the Weihrauch lattice. Our central results provide information on how Ramsey's theorem behaves under product, parallelization and jumps. From these results we can derive a number of important properties of Ramsey's theorem. For one, the parallelization of Ramsey's theorem for cardinality $n\geq1$ and an arbitrary finite number of colors $k\geq2$ is equivalent to the $n$--th jump of weak Kőnig's lemma. In particular, Ramsey's theorem for cardinality $n\geq1$ is $Σ^0_{n+2}$--measurable in the effective Borel hierarchy, but not $Σ^…
The rainbow Ramsey theorem states that every coloring of tuples where each color is used a bounded number of times has an infinite subdomain on which no color appears twice. The restriction of the statement to colorings over pairs (RRT22) admits several characterizations: it is equivalent to finding an infinite subset of a 2-random, to diagonalizing against Turing machines with the halting set as oracle... In this paper we study principles that are closely related to the rainbow Ramsey theorem, the Erdős Moser theorem and the thin set theorem within the framework of reverse mathematics. We pro…
We classify a sharp phase transition threshold for Friedman's finite adjacent Ramsey theorem. We extend the method for showing this result to two previously known classifications involving Ramsey theorem variants: the Paris--Harrington theorem and the Kanamori--McAloon theorem. We also provide tools to remove ad-hoc arguments from the proofs of phase transition results as much as currently possible.
We study four families of consequences of Ramsey's Theorem from the viewpoint of reverse mathematics. The first, which we call the Achromatic Ramsey Theorem, is from a partition relation introduced by Erdős, Hajnal and Rado: $ω\to [ω]^r_{c,\leq d}$, which asserts that for every $f: [ω]^r \to c$ there exists an infinite $H$ with $|f([H]^r)| \leq d$. The second and third are the Free Set Theorem and the Thin Set Theorem, which were introduced by Harvey Friedman. And the last is the Rainbow Ramsey Theorem. We show that, most theorems from these families are quite weak, i.e., they are strictly wea…
Inspired by Ramsey's theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinitely many vertices of $H$. We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramsey's theorem for pairs. We also identify a weak form of the Rival-Sands theorem th…
In this paper, we show that $\mathrm{RT}^{2}+\mathsf{WKL}_0$ is a $Π^{1}_{1}$-conservative extension of $\mathrm{B}Σ^0_3$.
In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $λ$, a strong coloring on $λ^+$ by $λ$ colors over a partition can be stretched to one with $λ^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds…
We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether $\RCA + \RRT^4_2 \vdash \ACA$ and obtain some partial answer.
Bounded Ramsey's theorem for triples in computability theory
研究三元组2染色的Ramsey定理受限形式，探讨其计算内容与配对Ramsey定理的相似性及差异。
In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from $\RT$, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below $\RT$ has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that $\ADS$ is not equivalent to $\CAC$ and that $\EM$ is not equivalent to $\RT$.
We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erd{ö}s and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various specia…
We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for $\ell$-colorings, $\mathsf{SRT}^2_\ell$. Our proof actually establishes the following considerably stronger fact: given $k > \ell$, there is a coloring $c : ω\to k$ such that for every stable coloring $d : [ω]^2 \to \ell$ (computable from $c$ or not), there is an infinite homogeneous set $H$ for $d$ that computes no infinite homogeneous set for $c$. This also answ…
A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák-Rödl …
Ramsey-like theorems and immunities
本文研究Ramsey-like定理的计算复杂性，证明其非计算平凡性，并探讨不同免疫概念下的定理特性。
I will give a presentation of an abstract approach to finite Ramsey theory found in an earlier paper of mine. I will prove from it a common generalization of Deuber's Ramsey theorem for regular trees and a recent Ramsey theorem of Jasinski for boron tree structures. This generalization appears to be new. I will also show, in exercises, how to deduce from it the Milliken Ramsey theorem for strong subtrees.
Ramsey's theorem for $n$-tuples and $k$-colors ($\mathsf{RT}^n_k$) asserts that every k-coloring of $[\mathbb{N}]^n$ admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its $Π^0_1$ consequences, and show that $\mathsf{RT}^2_2$ is $Π^0_3$ conservative over $\mathsf{I}Σ^0_1$. This strengthens the proof of Chong, Slaman and Yang that $\mathsf{RT}^2_2$ does not imply $\mathsf{I}Σ^0_2$, and shows that $\mathsf{RT}^2_2$ is finitistically reducible, in the sense of Simpson's partial realization of Hilbert's Pr…
The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a certain effective measure-theoretic sense. We show that the sets that can compute infinite homogeneous sets for non-null many computable stable colorings and the sets that can compute infinite homogeneous sets for all computable stable colorings agree below $\emp'$ but not in general. We also answer the analogs of two well known questions about the stable Ram…
Conservation of Ramsey's theorem for pairs and well-foundedness
研究证明对两个颜色的Ramsey定理在特定良基性公理下是Π1^1-保守的，改进了已有定理，推进了对偶Ramsey定理一阶部分的刻画。
Ramsey's theorem asserts that every $k$-coloring of $[ω]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable $k$-coloring of $[ω]^n$ whose solutions compute the halting set. On the other hand, for every computable $k$-coloring of $[ω]^2$ and every non-computable set $C$, there is an infinite monochromatic set $H$ such that $C \not \leq_T H$. The latter property is known as cone avoidance. In this article, we design a natural class of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal st…
On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC
本文在无选择公理的集合论框架下，考察广义 Rainbow Ramsey 定理与 Erdős–Rado 二元组 Canonical Ramsey 定理的强度，及其与多种弱选择原理的关系。
We show that although the Galvin-Prikry Theorem does not hold on generalized Baire space with the standard topology, there are similar theorems which do hold on generalized Baire space with certain coarser topologies.
Open questions about Ramsey-type statements in reverse mathematics
本文梳理逆数学及可计算、一致可归约性框架下研究 Ramsey 定理推论时遗留的开放问题，以暴露现有组合理解的空白。
The reverse mathematics of the Ordered Variable Word theorem
研究有序变量词定理的逆数学，证明其不蕴含对二染色对的拉姆齐定理，且每个可计算实例有低2度解，进而证明卡尔森-辛普森引理和开对偶拉姆齐定理在ACA0中可证。
Cross-constraint basis theorems and products of partitions
本文综述并扩展了陆刘的技术，证明了n元组三色Ramsey定理不能可计算归约到n元组两色Ramsey定理的有限乘积。
We analyze the Dual Ramsey Theorem for $k$ partitions and $\ell$ colors ($\mathsf{DRT}^k_\ell$) in the context of reverse math, effective analysis, and strong reductions. Over $\mathsf{RCA}_0$, the Dual Ramsey Theorem stated for Baire colorings is equivalent to the statement for clopen colorings and to a purely combinatorial theorem $\mathsf{cDRT}^k_\ell$. When the theorem is stated for Borel colorings and $k\geq 3$, the resulting principles are essentially relativizations of $\mathsf{cDRT}^k_\ell$. For each $α$, there is a computable Borel code for a $Δ^0_α$ coloring such that any partition h…
We show that RT(2,4) cannot be proved with one typical application of RT(2,2) in an intuitionistic extension of RCA0 to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach's axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility.
In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite $Π^0_2$ homogeneous set. The proof uses a countable collection of $Π^0_2$ sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of $Π^0_2$ sets. We provide a proof of this latter fact.
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a low_n solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdos-Moser theorem and stable Ramsey's theorem for pairs, while controlling their it…
A complete analysis is given of the computable reductions that hold between $\mathsf{SRT}^2_2$, $\mathsf{SPT}^2_2$, and $\mathsf{SIPT}^2_2$. In particular, while $\mathsf{D}^2_2\le_{\rm sW}\mathsf{SIPT}^2_2\le_{\rm sW}\mathsf{SPT}^2_2\le_{\rm sW}\mathsf{SRT}^2_2$, it is shown that $\mathsf{SRT}^2_2\not\le_{\rm sc}\mathsf{SPT}^2_2\not\le_{\rm sc}\mathsf{SIPT}^2_2\not\le_{\rm sc}\mathsf{D}^2_2$.
We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers $B$ on $ω$ as the prototype structures, we construct a class of continuum many topological Ramsey spaces $\mathcal{E}_B$ which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projection. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces $\mathcal{E}_B$, extending the Pudlak-Rodl Theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterati…
$Π^0_4$ conservation of Ramsey's theorem for pairs
该研究证明二元二色拉姆齐定理是$\u200a\mathsf{RCA}_0 + \mathsf{B}Σ^0_2\u200a$的$\u200a\forall Π^0_4\u200a$保守扩张，改进了已有结果，推进了二元拉姆齐定理一阶部分相关问题的解决。
We define a collection of topological Ramsey spaces consisting of equivalence relations on $ω$ with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of $ω$. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) i…
Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdos and Fred Galvin proved that for each coloring f, there is an infinite set that is "packed together" which is given "a small number" of colors by f. We analyze the strength of this theorem from the perspective of computability theory and reverse mathematics. We show that this theorem is close in computational strength to standard Ramsey's theorem by giving arithmetical upper and lower bounds for solutions to computable instances. In reverse mathematics, we show th…
We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales--Jewett theorems (for both located and nonlocated words), and the Farah--Hindman--McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken--Taylor theorem of Bergelson--Hindman--Williams as a particular case. We present applications of our Rams…
We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti by showing that every computable stable $2$-coloring of pairs has an incomplete $Δ^0_2$ infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT$^2_2$ does not imply RT$^2_2$.
Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdos and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type eta, where eta is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey's theorem for pairs. This Erdos-Rado theorem, like the tree theorem for pairs, belongs to a…
Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper developes a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to …
Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets. We also study the impact of the number …
Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.
Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic
本文在BΣ^0_3内构建了RT^2的\ll^2-解，并证明了其基理定理，同时展示了RT^2_2和EM_{<\infty}的类似定理，为相关保守性结果提供了更简单的证明。
A Topological Rainbow Ramsey Theorem
该研究在大基数存在条件下，证明了对于任意可数到一的着色，存在一个序类型为ω₁的闭集，使其在该集上着色为单射。该定理强化了两个已有定理并回答了Garti和Zhang的问题。
Ramsey's coheirs
本文利用模型论中的余继承者概念，简洁地证明了Ramsey理论中的多个经典定理，包括Ramsey定理、Hindman定理、Hales-Jewett定理，以及蕴含Carlson和Gowers分割定理的两个原理。
The cohesive and stable Ramsey theorems and proof size over a weak base theory
研究黏性Ramsey定理对偶数对的证明规模，展示其在弱基础理论RCA*上导致指数闭包，并探讨其对Π1句子的非初等加速及在RCA*+CAC中不可证性。
Carlson-Simpson's lemma and applications in reverse mathematics
本文研究 Carlson-Simpson 引理在逆向数学中的应用，证明 Open Dual Ramsey 定理在 ACA+0 中成立，并探讨其在 Ramsey 理论中的多种应用。
Recently, Solecki introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman's theorem, Carlson's theorem, and Gowers' FIN$_k$ theorem. He proved that an entire class of finite monoids is Ramsey. Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids. We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey Theorem. The results obtained suggest a possible connection with Schützenberger's theorem and finite automa…
We show a short proof of Higman's lemma using Friedman's adjacent Ramsey theorem for pairs. This provides an alternative proof of the known upper bound for the reverse mathematical status of Higman's lemma and that of its miniaturised version.
The reverse mathematics of bounded Ramsey's theorem for pairs
研究退化版Ramsey定理对二元对和两色（BRT²₂）的逆数学强度，证明其可由Erdős-Moser定理推出但不蕴含升降序列原理，且其可计算真版本等价于IΣ₂⁰。
Big Ramsey combinatorics of the Cantor set and a simple proof of Blass' perfect set theorem
本文通过研究康托尔集的大Ramsey组合学，证明了其有限的大Ramsey度，并给出了Blass完美集定理的简单证明，同时展示了完整布尔代数的有限大Ramsey度。
The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of $k$-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's Theorem for pairs and recursive assignments of $k$ colors is equivalent to the Limited Lesser Principle of Omniscience for $Σ^0_3$ formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite $k$-ary tree there is some $i < k$ and some branch with infinitely many childr…
We use Gödel's Dialectica interpretation to produce a computational version of the well known proof of Ramsey's theorem by Erdős and Rado. Our proof makes use of the product of selection functions, which forms an intuitive alternative to Spector's bar recursion when interpreting proofs in analysis. This case study is another instance of the application of proof theoretic techniques in mathematics.
We complete a 40-year old program on the computability-theoretic analysis of Ramsey's theorem, starting with Jockusch in 1972, and improving a result of Chong, Slaman and Yang in 2014. Given a set $X$, let $[X]^n$ be the collection of all $n$-element subsets of $X$. Ramsey's theorem for $n$-tuples asserts the existence, for every finite coloring of $[ω]^n$, of an infinite set $X \subseteq ω$ such that $[X]^n$ is monochromatic. The meta-mathematical study of Ramsey has a rich history, with several long-standing open problems and seminal theorems, including Seetapun's theorem in 1995 and Liu's t…
The paper is devoted to a reverse-mathematical study of some well-known consequences of Ramsey's theorem for pairs, focused on the chain-antichain principle $\mathsf{CAC}$, the ascending-descending sequence principle $\mathsf{ADS}$, and the Cohesive Ramsey Theorem for pairs $\mathsf{CRT}^2_2$. We study these principles over the base theory $\mathsf{RCA}^*_0$, which is weaker than the usual base theory $\mathsf{RCA}_0$ considered in reverse mathematics in that it allows only $Δ^0_1$-induction as opposed to $Σ^0_1$-induction. In $\mathsf{RCA}^*_0$, it may happen that an unbounded subset of $\mat…
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