Jaroslav Nešetřil, Vojtěch Rödl
We prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that class of acyclic graphs has the Ramsey property and uses the partite construction.
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Big Ramsey degrees of finite structures are usually considered with respect to a Fraï ssé limit. Building mainly on the work of Devlin, Sauer, Laflamme and Van Thé, in this paper we consider structures which are not Fraï ssé limits, and still have the property that their finite substructures have finite big Ramsey degrees in them. For example, the class of all finite acyclic oriented graphs is not a Fraï ssé class, and yet we show that there is a countably infinite acyclic oriented graph in which every finite acyclic oriented graph has finite big Ramsey degree. Our main tools come from categor…
Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $Δ\geq 2$ and any sufficiently large $n$, there exists an acyclic digraph…
Given an acyclic oriented graph $\vec{H}$ and a graph $G$, we write $G \to \vec{H}$ if every orientation of $G$ has an oriented copy of $\vec{H}$. We define $\vec{R}(\vec{H})$ as the smallest number $n$ such that there exists a graph $G$ satisfying $G \to \vec{H}$. Denoting by $R(H)$ the classical Ramsey number of a graph $H$, we show that $\vec{R}(\vec{H}) \leq 2R(H)^{c \log^2 h}$ for every acyclic oriented graph $\vec{H}$ with $h$ vertices, where $H$ is its underlying undirected graph. We also study the threshold function for the event $\{G(n,p) \to \vec{H}\}$ in the binomial random graph $G…
Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $χ(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph are colored with $t$ colors in any manner, there is a monochromatic copy of $T$. We observe that $χ(T,t)$ is well defined and $$\left\lceil {R^r(T,t)-1\over r-1}\right \rceil +1 \le χ(T,t)\le |E(T)|^t+1$$ where $R^r(T,t)$ is the $t$-color Ramsey number of $H$. We give linear upper bounds for $χ(T,t)$ when T is a matching or star, proving that for $r\ge 2, k\ge 1,…
We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.
We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.
Ramsey numbers of digraphs with local edge structure
该研究针对有向图的Ramsey数线性性问题，提出分级带宽刻画局部边结构，证明有界度无环且分级带宽有界的有向图定向Ramsey数线性，统一推广了多项已有结果。
If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{\vec H}=p_{\vec H}(n)$ for the property $G(n,p)\to \vec H$ for the cases in which $\vec H$ is an acyclic orientation of a complete graph or of a cycle.
We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Sokić. As a bonus, our proof gives new arguments for these two results.
For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or \emph{comparability graph} of $A$. Further, we say that a graph $G$ is a partial order graph if there exists a partially ordered set $A$ such that $G = G_A$. For a class $\mathcal{C}$ of simple, undirected graphs and $n$, $m \ge 1$, we define the Ramsey number $\mathcal{R}_{\mathcal{C}}(m,n)$ with respect to $\mathcal{C}$ to be the minimal number of vertices $r$…
Constrained Ramsey Numbers
对于两个图S和T，约束Ramsey数f（S，T）是最小的n，使得n个顶点上的完全图的任意颜色数的边染色都有一个与S同构的单色子图或一个与T同构的彩虹子图（所有边的颜色不同）. Erdos-Rado正则Ramsey定理暗示f（S，T）存在当且仅当S是星或T是无圈的，并且已经做了很多工作来确定f（S，T）对各种类型参数的增长率。当S和T都是分别有s和t条边的树时，Jamison，Jiang，and Ling证明了f（S，T）<= O（st^2），并证明了它总是至多O（st）。本文研究了T是一条路的情形，证明了f（S，P_t）= O（st log t），这与猜想仅相差一个对数因子。这实质上改进了大多数s和t值的先前边界。
In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Nešetřil and V. Rödl claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 M. Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of S. Solecki and M. Zhao that the class of all finite posets with several linear extensions has the Ramsey property. Using the categorical reinterpretation of the Ra…
On Ramsey number of Steiner systems
该研究证明了存在一个部分(k,k-1)-系统，其Ramsey数在r≥4种颜色下以高度为k-1的塔函数增长。
In this note we consider a Ramsey type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao.
We give an abstract approach to finite Ramsey theory and prove a general Ramsey-type theorem. We deduce from it a self-dual Ramsey theorem, which is a new result naturally generalizing both the classical Ramsey theorem and the dual Ramsey theorem of Graham and Rothschild. In fact, we recover the pure finite Ramsey theory from our general Ramsey-type result in the sense that the classical Ramsey theorem, the Hales--Jewett theorem (with Shelah's bounds), the Graham--Rothschild theorem, the versions of these results for partial rigid surjections due to Voigt, and the new self-dual Ramsey theorem …
The Cycle-Complete graph Ramsey numbers
当圈的阶数至少是完全图的阶数的4倍时，我们得到了圈与完全图的Ramsey数.这部分证实了Erdés、Faudree、Rousseau和Schelp在1978年提出的一个猜想。
We survey recent developments concerning two properties of classes of finite structures: the Ramsey property and the extension property for partial automorphisms (EPPA).
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals).
For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G, let the Ramsey number R_r(B(G), k) be the smallest integer n such that no matter how the edges of a complete r-uniform n-vertex hypergraph are colored with k colors, there is a copy of a monochromatic Berge-G subhypergraph. Furthermore, let R(B(G),k) be the smallest integer n such that no matter how all subsets an n-element set are colored with k colors, th…
Corrigendum and Addendum to "Fra\"ıssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition''
本文更正了原论文中关于某个泛许排列的reduct误认为是泛序的错误，并展示了该reduct实际上是二维泛序，同时指出该修正带来了新的研究方向，即证明了一类未被探索的泛关系结构具有有限的big Ramsey度数。
We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures,extending Zheng's work for the profinite graph to the setting of Fra\"ıssé classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-Läuchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fra\"ıssé class, its universal inverse limit structure has finite big Ramse…
Characterisation of the big Ramsey degrees of the generic partial order
该研究通过改进上界方法、提出下界新技术，刻画了通用偏序的大拉姆齐度，实现了有限偏序拉姆齐类结论的无穷扩展。
The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions and NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. Proper interval graphs are precisely those which can be oriented as acyclic local tourn…
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Nešetřil-Rödl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results s…
Proper forcing and rectangular Ramsey theorems
我证明强制产品的可定义的偏序保持共尾性的微薄或零理想的保存定理。矩形拉姆齐定理相关的理想遵循的证明。
Multicolor Ramsey numbers of ordered matchings
本文否定回答了有序匹配多色Ramsey数的多项式对数上界猜想，证明几乎每个完美匹配的下界指数为$q$次对数，并给出两个应用：证明有界度无环有向图的下界猜想和加强$M$-自由有序图色数下界。
Given graphs $ F_1, F_2$ and $G$, we say that $G$ is Ramsey for $(F_1,F_2)$ and we write $G\rightarrow(F_1, F_2)$, if for every edge coloring of $G$ by red and blue, there is either a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The size Ramsey number $\hat{r}(F_1, F_2)$ is defined as the minimum number of edges of a graph $G$ such that $G\rightarrow(F_1, F_2)$. This paper provides the exact value of $\hat{r}(F_1, F_2)$ for many pairs $(F_1, F_2)$ of star forests, giving a partial solution to a conjecture of Burr et al. (Ramsey-minimal graphs for multiple copies, Indagationes Mathematicae…
Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a $σ$-closed `almost reduction' relation analogously to the partial ordering of `mod finite' on $[ω]^ω$. Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler a…
We prove a generalization of Gowers' theorem for $\mathrm{FIN}_{k}$ where, instead of the single tetris operation $T:\mathrm{FIN}_{k}\rightarrow \mathrm{FIN}_{k-1}$, one considers all maps from $\mathrm{FIN}_{k}$ to $\mathrm{FIN}_{j}$ for $0\leq j\leq k$ arising from nondecreasing surjections $f:\left\{ 0,1,\ldots ,k+1\right\} \rightarrow \left\{ 0,1,\ldots ,j+1\right\} $. This answers a question of Bartošová and Kwiatkowska. We also prove a common generalization of such a result and the Galvin--Glazer--Hindman theorem on finite products, in the setting of layered partial semigroups introduced…
Volume inequalities for flow polytopes of full directed acyclic graphs
本文研究固定度序列的有向无环图的流多胞体体积，证明存在与布尔代数同构的偏序，沿该偏序体积弱递减，并给出二分非交叉树的线性扩展解释。
The class of finite distributive lattices, as many other classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokić have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the variety of distribu…
We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$ vertices contains either a rainbow triangle or a monochromatic copy of $G$. It turns out that $gr_k(K_3, G)$ behaves more nicely than the classical Ramsey number $r_k(G)$. However, finding exact values of $gr_k (K_3, G)$ is far from trivial. In this paper, we prove that $gr_k(K_3, C_9)= 4\cdot 2^k+1$ for all $k\ge1$. This new result provides partial evidence fo…
Let $G$ be a graph. We introduce the acyclic b-chromatic number of $G$ as an analogue to the b-chromatic number of $G$. An acyclic coloring of a graph $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the induced subgraph on vertices of any two colors $i,j\in \{1,\dots,k\}$ induces a forest. On the set of all acyclic colorings of $G$ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number $A(G)$ of $G$ and the maximum cardinality of its minimal el…
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. We study Ramsey-type problems in Gallai colorings. Given an integer $k\ge1$ and a graph $H$, the Gallai-Ramsey number $GR_k(H)$ is the least positive integer $n$ such that every Gallai $k$-coloring of the complete graph on $n$ vertices contains a monochromatic copy of $H$. It turns out that $GR_k(H)$ is more well-behaved than the classical Ramsey number $R_k(H)$. However, finding exact values of $GR_k (H)$ is far from trivial. In this …
Acyclic heaps of pieces, I
堆件介绍了Viennot和应用代数组合，理论计算机科学和统计物理。在本文中，我们如何某些组合性质的范和Stembridge研究堆是密切相关的性质，一定的线性映射$\partial_E$相关联的堆$E$。当$F$是$E$的一个子堆时，我们研究了$\partial_E$和$\partial_F$之间的关系。这种方法允许整洁的声明和证明某些结合代数（广义Temperley-李伯代数）的结果，否则很难证明。证明的关键是用映射$\partial$来解释上述代数的结构常数。
Ramsey Obstructions to Disambiguation
本文构造部分矩阵，其每个去歧义化都包含任意给定大小的二元矩阵，利用拉姆齐理论回答线性分类器去歧义化问题，并给出部分图对称结果。
Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of its edges contains a red copy of $G$ or a blue copy of $H$. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes th…
Hamiltonicity of graphs of acyclic orientations and acyclic polynomials
该研究刻画了$\text{AO}$-哈密顿的多路径，提出保持$\text{AO}$-哈密顿性的粘合准则与方法，定义无环多项式并探究其性质，为$\text{AO}$-哈密顿性的归纳认证提供支撑。
A note on degree conditions for Ramsey goodness of trees
本文改进了树的Ramsey优良度条件，将路径推广为任意树并优化了度数下界。
We introduce polytopal cell complexes associated with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, two of these polytopal cell complexes are observed to minimally resolve certain special combinatorial polynomial ideals related to acyclic orientations. These ideals are explicitly found to be Alexander dual, which relative to comparable results in the literature, generalizes in a cleaner and more illuminating way the well-known duality between permutohedron and tree ideals. The combinatorics underlying these resu…
An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$
本文提出了 $r$-一致完全超图 $K_{rd}^{(r)}$ 的无环齐次 $d$-划分，并以此证明映射 $det^{S^r}$ 对所有 $r$ 均非平凡，部分回答了相关猜想。
A positive definite completion problem pertains to determining whether the unspecified positions of a partial (or incomplete) matrix can be completed in a desired subclass of positive definite matrices. In this paper we study an important and new class of positive definite completion problems where the desired subclasses are the spaces of covariance and inverse-covariance matrices of probabilistic models corresponding to directed acyclic graph models (also known as Bayesian networks). We provide fast procedures that determine whether a partial matrix can be completed in either of these spaces …
We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil-Rödl Theorem, the Ramsey property of partial orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered $S$-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the str…
Online Ramsey numbers of ordered paths and cycles
本文研究固定有序图G时网络有序拉姆齐数r_o(G,P_n)的增长行为，证明了一般情形下的O(n log₂ n)上界及G为3-色图时的O(n)上界，并部分分类了满足r_o(G,P_n)=n+O(1)的图。
Twenty years of Nešetřil's classification programme of Ramsey classes
本文回顾了Nešetřil提出的Ramsey类分类计划在过去二十年的进展，列出了未解决的问题，并讨论了向EPPA和大Ramsey结构等新领域的扩展。
Big Ramsey degrees using parameter spaces
本文利用参数空间和Carlson-Simpson定理证明泛齐次偏序具有有限大拉姆齐度，并将该技术应用于三角形无关图、有限距离度量空间及Urysohn球面的振荡稳定性。
Monochromatic graph decompositions inspired by anti-Ramsey colorings
该研究推广反拉姆齐/彩虹染色问题，定义遗传图族下的染色阈值函数与相关参数，证明阈值的渐近公式，并通过示例说明其应用，拓展了图染色理论的研究范围。
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain model-theoretic structures. In 2005, Nešetřil initiated a systematic study of the so-called Ramsey classes of finite structures. This thesis is a contribution to the programme; we find Ramsey expansions of the primitive 3-constrained classes from Cherlin's catalogue of metrically homogeneous graphs. A key ingradient is an explicit combinatorial algorithm to fill-in…
We investigate Ramsey expansions, the coherent extension property for partial isometries (EPPA), and the existence of a stationary independence relation for all classes of metrically homogeneous graphs from Cherlin's catalogue. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and coherent EPPA. Our results are a contribution to Nešetřil's classification programme of Ramsey classes and c…
Multicolor Ramsey numbers on stars versus pat
本文研究多颗星形图与一条路径图的多色 Ramsey 数，在特定模同余条件下给出新下界与精确值，部分证实 Wang 的猜想。
Let $G$ be a simple graph and let $\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\textbf k$-chromatic polynomial of $G$ in terms of dimensions of the grade spaces of $\mathcal{L}(G)$. This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of $G$ are obtained as corollaries. A heap with a unique minimal piece is said to be a …
Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is $2^gT(3,1/2)$. We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is $2^{n-1}T(1/2,3)$. We then prove that the number of partial orientations modulo cycle reversals is $2^gT(3,1)$ and the number of partial orientations modulo cut reversals is $2^{n-1}T(1,3)$. To prove these results, we introduce cut and cycle minimal pa…
Some polynomial classes for the acyclic orientation with parity constraint problem
本文研究了无环定向问题中的奇偶约束，提出多项式类以确定无环T-奇定向的存在条件，并通过构造性证明展示了其在特定图结构中的可解性。
We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan's cycle--cocycle reversal system for partial orientations. An oriented version of Dhar's burning algorithm is introduced and employed in the study of acyclicity for partial orientations. We then show that the Baker--Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle--cocycle reversal system to produce an acyclic partial orientation. These results are applied in providing new proo…
We introduce the problem of Weak Dominance Drawing for general directed acyclic graphs and we show the connection with the linear extension diameter of a partial order P. We present complexity results and bounds.
The queue-number of a poset is the queue-number of its cover graph viewed as a directed acyclic graph, i.e., when the vertex order must be a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue-number at most $w$. Recently, Alam et al. constructed posets of width $w$ with queue-number $w+1$. Our contribution is a construction of posets with width $w$ with queue-number $Ω(w^2)$. This asymptotically matches the known upper bound.
Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs
本文提出在加权有向无环多图上扩展莫比乌斯变换和沙普利值，以处理向量值函数，引入投影算子和新公理，实现对复杂系统结构的分解与归因。
In this paper we prove a general theorem showing the extension property for partial automorphisms (EPPA, also called the Hrushovski property) for classes of structures containing relations and unary functions, optionally equipped with a permutation group of the language. The proof is elementary, combinatorial and fully self-contained. Our result is a common strengthening of the Herwig-Lascar theorem on EPPA for relational classes with forbidden homomorphisms, the Hodkinson-Otto theorem on EPPA for relational free amalgamation classes, its strengthening for unary functions by Evans, Hubička and…
We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures fall into precisely five equivalence classes. We achieve this result by showing that there exist exactly five closed permutation groups which contain the automorphism group of the random partial order, and thus expose all symmetries of this structure. Our classification lines up with previous similar classifications, such as the structures definable in the random graph or the order of the rationals; it also provides…
We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.
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