Csaba Biró, Sida Wan
Let $\mathbf{k}$ denote the totally ordered set (or chain) on $k$ elements. The product $\mathbf{k}^t=\mathbf{k}\times\cdots\times\mathbf{k}$ is a poset called a grid. This paper discusses several loosely related results on the Ramsey theory of grids. Most of the results involve some application of the Product Ramsey Theorem.
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In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than $ω^ω$. Big Ramsey degrees of finite chains in all other countable ordinals are infinite.
Motivated by the paper of Axenovich and Walzer [2], we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring on elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also give some general upper and lo…
A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite sums have finite big Ramsey spectra. Since big Ramsey spectra of all non-scattered countable chains are finite by results of Galvin, Laver and Devlin, in order to complete the characterization of …
A short note on the characterization of countable chains with finite big Ramsey spectra
该研究证实可数散布链的复杂性与其大拉姆齐组合学的结构对应，证明其有有限大拉姆齐度等价于有限豪斯多夫秩，还将大拉姆齐谱概念推广到单态结构并给出充分条件。
Products of simplices are canonically Ramsey
研究单纯形的乘积是否为典范拉姆齐，通过扩展前人成果，证明所有单纯形的乘积都是典范拉姆齐的。
Ramsey numbers for partially-ordered sets
本文研究了t-链的弱和强偏序拉姆齐数，给出了相关数的上下界。
One-point extensions of Euclidean Ramsey sets
该研究证明有限欧氏拉姆齐集X的仿射包外添加任意点仍为欧氏拉姆齐集，解答了Ivan等人的猜想，给出了分情形的严格证明。
Poset Ramsey number $R(P,Q_n)$. III. Chain Compositions and Antichains
本文研究偏序集 Ramsey 数 $R(P,Q_n)$，为链的平行复合和细分 $Q_2$ 两类偏序集给出上界，从而完成至多4元素偏序集的结果确定，并证明反链情形 $R(A_t,Q_n)=n+3$ 对 $3\le t\le \log\log n$ 成立。
Big Ramsey degrees and the two-branching pseudotree
本文研究了伪树中有限链的大 Ramsey 度，证明其有限且为七，与反链的无限度形成对比，揭示了伪树中不同子结构的大 Ramsey 度差异。
A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q, and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way. Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q, each of which has a syst…
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.
In this note we study Shannon capacity of channels in the context of classical Ramsey numbers. We overview some of the results on capacity of noisy channels modelled by graphs, and how some constructions may contribute to our knowledge of this capacity. We present an improvement to the constructions by Abbott and Song and thus establish new lower bounds for a special type of multicolor Ramsey numbers. We prove that our construction implies that the supremum of the Shannon capacity over all graphs with independence number 2 cannot be achieved by any finite graph power. This can be generalized t…
Ramsey algebras is an attempt to investigate Ramsey spaces generated by algebras in a purely combinatorial fashion. Previous studies have focused on the basic properties of Ramsey algebras and the study of a few specific examples. In this article, we study the properties of Ramsey algebras from a structural point of view. For instance, we will see that isomorphic algebras have the same Ramsey algebraic properties, but elementarily equivalent algebras need not be so, as expected. Answer to a long-standing question about the Cartesian products of Ramsey algebras is also given.
Ramsey-Turán theory for partially-ordered sets
引入弱和强偏序集Ramsey-Turán数，研究布尔格族中t-链的极值问题，给出精确值、通用上界及渐近下界。
For a positive integer $r$, let $G(r)$ be the smallest $N$ such that, whenever the edges of the Cartesian product $K_N \times K_N$ are $r$-coloured, then there is a rectangle in which both pairs of opposite edges receive the same colour. In this paper, we improve the upper bounds on $G(r)$ by proving $G(r) \leq \Big(1 - \frac{1}{128}r^{-2}\Big) r^{\binom{r+1}{2}}$, for $r$ large enough. Unlike the previous improvements, which were based on bounds for the size of set systems with restricted intersection sizes, our proof is a form of a quasirandomness argument.
W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace generated by that sequence is monochromatic. For $k=1,$ this immediately gives the finite sum theorem. In this article we will show that for any finite coloring of $X_{k}$ there exist two sequences $\left\{ \mathbf{n_{i}}:i\in I\right\} $ and $\left\{ \mathbf{m_{i}}:i\in I\right\} $ such that the Gower subspace generated by $\left\{ \mathbf{n_{i}}:i\in I\right\}…
The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product $K_n \times K_n$ in $r$ colors then, for $n$ sufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Shelah's proof shows that $n = r^{\binom{r+1}{2}} + 1$ suffices. More than tw…
Given posets $\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k$, let the {\em Boolean Ramsey number} $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ be the minimum number $n$ such that no matter how we color the elements in the Boolean lattice $\mathbf{B}_n$ with $k$ colors, there always exists a poset $\mathbf{P}_i$ contained in $\mathbf{B}_n$ whose elements are all colored with $i$. This function was first introduced by Axenovich and Walzer~\cite{AW}. Recently, many results on determining $R(\mathbf{B}_m,\mathbf{B}_n)$ have been published. In this paper, we will study the function $R(\mathbf…
Ramsey Theory for Product Trees
本文发展了乘积树的拉姆齐理论，证明了在足够大的树乘积中，大叶子子集的祖先闭包必含特定结构的算术分支副本，并推广到多叉树乘积，给出临界指数，几何上对应高维集合中的分支模式。
A combinatorial study on product of filter large sets
该研究针对拉姆齐理论中的大集乘积性质问题，证明两个沿幂等滤子的组合大集的乘积仍为滤子大集，且证明采用完全初等的技巧，丰富了滤子大集的组合理论结论。
Ramsey numbers of grid graphs
本文研究了网格图的非对角Ramsey数，证明了除$G_{2\times 2}$外的任意环及递归膨胀的网格子图，其Ramsey数均为多项式级别，并基于多色Erdős-Hajnal猜想给出了两行网格子图的多项式上界。
Subgroups of the symmetric group $S_n$ act on powers of chains $C^n$ by permuting coordinates, and induce automorphisms of the ordered sets $C^n$. The quotients defined are candidates for symmetric chain decompositions. We establish this for some families of groups in order to enlarge the collection of subgroups $G$ of the symmetric group $S_n$ for which the quotient $B_n/G$ obtained from the $G$-orbits on the Boolean lattice $B_n$ is a symmetric chain order. The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.
We present two results on maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\ldots\times[t_n+1]$. First, we prove that these maximal antichains are also maximum. Second, we prove that there is a bijection between maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\ldots\times[t_n+1]$ and antichains in the non-strict chain product poset $[t_1]\times[t_2]\times\ldots\times[t_n]$.
J. Propp and T. Roby isolated a phenomenon in which a statistic on a set has the same average value over any orbit as its global average, naming it homomesy. They proved that the cardinality statistic on order ideals of the product of two chains poset under rowmotion exhibits homomesy. In this paper, we prove an analogous result in the case of the product of three chains where one chain has two elements. In order to prove this result, we generalize from two to $n$ dimensions the recombination technique that D. Einstein and Propp developed to study homomesy. We see that our main homomesy result…
Flip Graphs on Self-Complementary Ideals of Chain Products
本文在链积偏序集的自补理想上引入翻转操作，研究了所得翻转图的顶点数渐近估计、直径和半径界，并在满足额外对称性的链积上获得了类似结果。
We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the Möbius function of the Rees product of any graded poset with the chain, and more generally, the t-ary tree, is exactly the same as the Rees product of its dual with the chain, respectively, t-ary chain. We then study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give a bijective proof that the Möbius function of this poset can be expressed as n times a signed derangement number. From …
We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is "taut", i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions --- the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symme…
We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of nonzero dot products $(α_1,\ldots,α_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(A_1,\dots, A_{k+1})\in E^{k+1}$ satisfying $A_j \cdot A_{j+1}=α_j$ for every $1\leq j \leq k$.
We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset. In a product of chains, the maximum size of a chain that Maker can guarantee building is $k-\lfloor r/2\rfloor$, where $k$ is the maximum size of a chain in the product, and $r$ is the maximum size of a factor chain. We also study a variant in which Maker must follow the chain in order, called the {\it Walker-Blocker game}. In the poset consisting of the bottom $k$ levels of the product of $d$ arbitrarily long chains, Walker can guarantee a chain that hits all levels if $d\ge14$; this result uses a solution t…
Interval-closed set rowmotion and homomesy on products of two chains
本文研究区间闭集上的行移动动力学，简化了其全局定义，完整刻画了两条链乘积上的轨道结构，并据此证明了关于带符号基数统计量的同稳性猜想。
The magic of tensor products of ultrafilters
该研究探讨超滤子张量积的组合性质，证明其新特征、相关组合结构结果及半群幂等元描述，拓展了其在数论等领域的应用。
An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of non-linear patterns which can be found in a single cell of any finite partition of $\mathbb{N}$. Our proof involves a correspondence principle which transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.
We define a finite Markov chain, called generalized crested product, which naturally appears as a generalization of the first crested product of Markov chains. A complete spectral analysis is developed and the $k$-step transition probability is given. It is important to remark that this Markov chain describes a more general version of the classical Ehrenfest diffusion model. As a particular case, one gets a generalization of the classical Insect Markov chain defined on the ultrametric space. Finally, an interpretation in terms of representation group theory is given, by showing the corresponde…
Tensor product Markov chains and Weil representations
本文研究了有限经典群不可约表示上由Weil表示张量积诱导的马尔可夫链，并给出了其收敛速度的尖锐界限。
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