M. Malliaris, S. Shelah
This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.
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Szemerédi's regularity lemma is a powerful tool in graph theory. It states that for every large enough graph, there exists a partition of the edge set with bounded size such that most induced subgraphs are quasirandom. When the graph is a definable set $ϕ(x, y)$ in a finite field $F_q$, Tao's algebraic graph regularity lemma shows that there is a partition of the graph $ϕ(x, y)$ such that all induced subgraphs are quasirandom and the error bound on quasirandomness is $O(q^{-1/4})$. In this work we prove an algebraic hypergraph regularity lemma for definable sets in finite fields, thus answerin…
In a recent paper, Chernikov and Starchenko prove that graphs defined in distal theories have strong regularity properties, generalizing previous results about graphs defined by semi-algebraic relations. We give a shorter, purely model-theoretic proof of this fact.
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-…
On the regularity of almost stable relations
研究属于理想（如测度零）的局部稳定性，证明了几乎稳定公式的平稳性原理，构建了 Cantor-Bendixson 秩有限的拓扑空间，并应用于无限图和有限群的正则性引理。
Zarankiewicz bounds from distal regularity lemma
本文证明所有满足远距离正则性引理的关系均享有改进的扎兰基维茨界，扩展了半代数关系的上界研究。
We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and the formula R(x,y) is stable. The proof is rather quick and uses local stability theory. The special case where (V,W,R) is pseudofinite, mu, nu are the counting measures and M is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of Malliaris-Shelah (Transactions AMS, 366, 2014, 1551-1585), though without explicit bounds or equitability.
Our work builds on known results for k-uniform hypergraphs including the existence of limits, a Regularity Lemma and a Removal Lemma. Our main tool here is a theory of measures on ultraproduct spaces which establishes a correspondence between ultraproduct spaces and Euclidean spaces. First we show the existence of a limit object for convergent sequences of relational structures and as a special case, we retrieve the known limits for graphs and digraphs. Then we extend this notion to finite models of a fixed universal theory. We also state and prove a Regularity Lemma and a Removal Lemma. We wi…
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 proper…
Let G be a finite graph with the non-k-order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). A major result of the paper applies model-theoretic arguments to obtain a stronger version of Szemerédi's regularity lemma for such graphs, Theorem 5.18, in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition. Motivation for this work comes from a coincidence of model-theoretic and graph-theoretic ideas. Namely, it was known that the "irregular pairs" in the statement of Szemerédi's …
Homogeneous hypergraph regularity lemmas via $k$-strong honest definitions
研究在NIP强k-远结构中可定义的(k + 1)-一致超图，通过引入k-强诚实定义，证明其满足齐次正则引理，扩展了Chernikov和Starchenko的相关成果，将强诚实定义理论推广到高元情形。
Pseudofinite proofs of the stable graph regularity lemma
本文基于讲座，用局部稳定性理论和超积构造简化证明了稳定图正则引理，并通过函数误差加强结果，还给出提取等划分的初等论证。
An arithmetic algebraic regularity lemma
本文提出一个算术规则引理，用于有限域中可定义群的规则性分析，证明了群的正规子群具有特定的随机性质。
The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in 2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$ of bounded codimension such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. It is known that in general, the growth of the codimension of $H$ is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non-uniform cosets. Our main result is that, under a natural model-theoretic assumption of stability, the tower-type bound and non-unifor…
The ineffectiveness of the regularity lemma for bounded degree graphs
本文证明对于任意 $Δ\geq 3$，不存在可从 $(\varepsilon, r)$ 计算的界来限定以 $\varepsilon$ 误差近似最大度为 $Δ$ 的图所需图的大小，从而否定了 Lovász 的相关问题，此结论源于反驳 Aldous-Lyons 猜想的近期工作。
Perfect stable regularity lemma and slice-wise stable hypergraphs
本文研究了超图的模型论稳定性及其对顶点划分的正则性引理加强，给出了三元情形的完整分类，并构造反例回应了Terry和Wolf的猜想，结果适用于有限图超积与可定义Keisler测度。
In this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.
Quantitative analytic stable regularity
研究二元实值函数的定量稳定正则性，通过“解析对称引理”等工具证明相关引理，并对随机抽样方法进行函数理论处理，扩展了Malliaris和Shelah关于稳定图的工作。
Stabilizers and NIP arithmetic regularity
该研究为有限群NIP算术正则引理提供了无需复杂模型论工具的初等新证明，拓展了相关稳定子引理，还证明了NIP集合满足强形式多项式Bogolyubov-Ruzsa猜想。
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