Andrii Arman, Sergei Tsaturian
It is proved that if the points of the three-dimensional Euclidean space are coloured in red and blue, then there exist either two red points unit distance apart, or six collinear blue points with distance one between any two consecutive points.
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In this paper, we introduce Euclidean Gallai-Ramsey theory, by combining Euclidean Ramsey theory and Gallai-Ramsey theory on graphs. More precisely, we consider the following problem: For an integer $r$ and configurations $K$ and $K'$, does there exist an integer $n_0$ such that for any $r$-coloring of the points of $n$-dimensional Euclidean space with $n \geq n_0$, there is a monochromatic configuration congruent to $K$ or a rainbow configuration congruent to $K'$? In particular, we give a bound on $n_0$ for some configurations $K$ and $K'$, such as triangles and rectangles. Those are extensi…
We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdos distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3,k).
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。
For fixed graphs $F_1,\dotsc,F_r$, we prove an upper bound on the threshold function for the property that $G(n,p) \to (F_1,\dotsc,F_r)$. This establishes the $1$-statement of a conjecture of Kohayakawa and Kreuter.
It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it is improved to 13 <= N^* <= N.
Nearly all known Euclidean Ramsey sets are subsoluble
本文证明几乎所有已知的欧几里得雷姆赛集都是子可解的，仅存在两个可能的例外。
Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points at distance one. Answering a question raised by Fox and the first author and independently by Arman and Tsaturian, we show that there is a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ that contains no red copy of $\ell_3$ and no blue copy of $\ell_m$.
Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory
本文证明任意有限仿射无关集可等距嵌入正多边形环面，并借助 Kříž 定理给出所有有限仿射无关集均为 Ramsey 的新证明。
Progressions in Euclidean Ramsey theory
本研究改进了欧几里得拉姆齐理论中共线点着色问题的结果，将蓝色单位间距共线点的阈值从$10^{50}$降至1177，并拓展了不同点间距下的相关结论。
A finite set $X$ in some Euclidean space $R^n$ is called Ramsey if for any $k$ there is a $d$ such that whenever $R^d$ is $k$-coloured it contains a monochromatic set congruent to $X$. This notion was introduced by Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory. In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever `rival' c…
An old question in Euclidean Ramsey theory asks, if the points in the plane are red-blue coloured, does there always exist a red pair of points at unit distance or five blue points in line separated by unit distances? An elementary proof answers this question in affirmative.
Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by Łuczak that applies Szemerédi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain th…
Non-spherical sets versus lines in Euclidean Ramsey theory
本文证明了对任意非球面集 X，存在自然数 m 及对任意维数 n 的欧几里得空间着色，避免红色 X 副本和蓝色单位距离等差数列，验证了 Wu 与第一作者的猜想。
Asymmetric Ramsey properties of randomly perturbed graphs
本文研究稠密图加入随机边后的非对称 Ramsey 阈值，确定多类新图对与密度范围的结果，并解决 $(K_t,K_s)$ 在 $t\geq s\geq 5$ 时的遗留情形。
Bipartite and Euclidean Gallai-Ramsey Theory
本文研究二部图与欧几里得空间的两类Gallai-Ramsey问题，证明二部图情形答案关于色数线性并给出s=2时的精确界，还通过图着色与空间着色的对应得到一类构型的欧氏Gallai-Ramsey数新上界。
A graph $F$ is Ramsey for a pair of graphs $(G,H)$ if any red/blue-coloring of the edges of $F$ yields a copy of $G$ with all edges colored red or a copy of $H$ with all edges colored blue. Two pairs of graphs are called Ramsey equivalent if they have the same collection of Ramsey graphs. The symmetric setting, that is, the case $G=H$, received considerable attention. This led to the open question whether there are connected graphs $G$ and $G'$ such that $(G,G)$ and $(G',G')$ are Ramsey equivalent. We make progress on the asymmetric version of this question and identify several non-trivial fam…
On the use of senders for asymmetric tuples of cliques in Ramsey theory
本文针对任意团组构造了信号发送者和决定者工具图，并利用这些工具将该领域的三个经典定理推广至非对称多色 Ramsey 设置。
In the paper we prove, in particular, that for any measurable coloring of the euclidian plane into two colours there is a monochromatic triangle with some restrictions on the sides. Also we consider similar problems in finite fields settings.
Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ an…
It is proved that for $k\geq 4$, if the points of $k$-dimensional Euclidean space are coloured in red and blue, then there are either two red points distance one apart or $k+3$ blue collinear points with distance one between any two consecutive points. This result is new for $4\leq k\leq 10$.
We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always fin…
Euclidean Gallai-Ramsey for various configurations
本文研究着色欧几里得空间中多种构型的Gallai-Ramsey问题，涵盖三角形、正方形、直线及矩形和球面构型，改进已有结果并揭示空间维数与颜色数无关的现象。
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…
A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq cn^{-1/m_2(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the "0-statement"), while for $p \geq Cn^{-1/m_2(F)}$ it is $1-o(1)$ (the "1-statement"). Here $m_2(F)$ denotes the $2$-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is det…
A conjecture of Leader, Russell and Walters in Euclidean Ramsey theory says that a finite set is Ramsey if and only if it is congruent to a subset of a set whose symmetry group acts transitively. As they have shown the ``if" direction of their conjecture follows if all finite groups have a Hales--Jewett type property. In this paper, we show that this property is satisfied in the case of finite solvable groups. Our result can be used to recover the work of Kříž in Euclidean Ramsey theory.
Ramsey problems for graphs in Euclidean spaces and Cartesian powers
本文研究了欧几里得空间和笛卡尔幂中图的拉姆齐问题，提出了新的函数χ_H(R^n)，并探讨了其在不同图结构下的性质及应用。
Asymmetric Ramsey numbers of trees
该研究通过四种构造给出树的非对称Ramsey数的下界公式，并证明在最大度满足对数阶上界、二分划类大小及顶点数满足特定条件时，该下界即为精确值，同时给出反例说明三个假设的必要性。
We say that $G \to (F,H)$ if, in every edge colouring $c: E(G) \to \{1,2\}$, we can find either a $1$-coloured copy of $F$ or a $2$-coloured copy of $H$. The well-known Kohayakawa--Kreuter conjecture states that the threshold for the property $G(n,p) \to (F,H)$ is equal to $n^{-1/m_{2}(F,H)}$, where $m_{2}(F,H)$ is given by \[ m_{2}(F,H):= \max \left\{\dfrac{e(J)}{v(J)-2+1/m_2(H)} : J \subseteq F, e(J)\ge 1 \right\}. \] In this paper, we show the $0$-statement of the Kohayakawa--Kreuter conjecture for every pair of cycles and cliques.
Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory
研究欧几里得Ramsey理论中避免红色直线段和蓝色1-分离集所需的最小集大小，改进了下界至(11+o(1))^n ln R，并在特殊情形下达到(5+o(1))^n ln R，同时解决了Erdős和Graham关于二维空间的问题。
The threshold for the asymmetric vertex-Ramsey property in randomly perturbed graphs
研究随机扰动图中非对称顶点-Ramsey性质，确定需添加的随机边数，使得稠密图以高概率成为(H1,...,Hr)_v-Ramsey。
Graphs with asymmetric Ramsey properties
本文利用概率方法和超图容器证明存在图 $G$ 满足 $G \nrightarrow (K_k,K_k)$ 且 $G \rightarrow (K_{R(k)-1},K_{k-1})$，作为 Nešetřil 和 Rödl 经典定理的非对称变体。
Generalised Prisms and Euclidean Ramsey Theory
本文证明若有限等距群在集合X和Y上传递，则它们在R^{d+1}中形成的棱柱也包含在传递等距群的有限集中，且可解性保持，为生成拉姆齐集提供新工具。
According to a classical result of Szemerédi, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to Fürstenberg and Katznelson says that every dense subset of $\{1,2,...,N\}^d$ contains an arbitrary large grid, if $N$ is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval $[0,L]$ on the line contains an arbitrary long approximate arithmetic progression, if $L$ is large en…
Multiple Distance Ramsey Bounds For Graphs in Euclidean Spaces
该研究扩展了图副本框架与多重距离定理，证明了欧氏空间中图的多重距离拉姆齐下界，改进了部分图族的一般界，并构造了半对角压扁秩定理的加权强化版本。
In this paper we exhibit a similarity between Euclidean Ramsey problems and Knaster-type problems. By borrowing ideas from Ramsey theory we prove weak Knaster properties of non-equatorial triangles in spheres, and of simplices in Euclidean spaces.
A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,\ldots,H_q)$, denoted by $G\to_q(H_1,\ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $i\in[q]$. Let $s_q(H_1,\ldots,H_q)$ denote the smallest minimum degree of $G$ over all graphs $G$ that are minimal $q$-Ramsey for $(H_1,\ldots,H_q)$ (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, Erdős and Lovász, who determined its value precisely for a pair of cliques. Over the past two decades the parameter $s_q$ has been studie…
A note on hypergraphs with asymmetric Ramsey properties
本文研究了超图的非对称Ramsey性质，证明了对于任意整数t₁≥…≥tₗ>r，存在一个r-超图G，使得G不包含K^{(r)}_{t₁},…,K^{(r)}_{tₗ}的单色复制，但包含K^{(r)}_s,K^{(r)}_{tₗ-1}的单色复制，其中s=R(K^{(r)}_{t₁},…,K^{(r)}_{tₗ})-1。
Block Sizes in the Block Sets Conjecture
研究块集猜想中块大小的界限，证明对于某些模板，块大小无法被限制，且对于模板123，块大小可为2，优于以往的塔型上界。
Avoiding short progressions in Euclidean Ramsey theory
本文提出构造欧几里得空间双色及三色着色的通用框架，显著改进避免单色共线等距点构型的界限，并肯定回答 Führer 与 Tóth 的问题。
We recursively extend the Lovász theta number to geometric hypergraphs on the unit sphere and on Euclidean space, obtaining an upper bound for the independence ratio of these hypergraphs. As an application we reprove a result in Euclidean Ramsey theory in the measurable setting, namely that every $k$-simplex is exponentially Ramsey, and we improve existing bounds for the base of the exponential.
Nowhere dense Ramsey sets
本文针对拉姆齐集，构造了无限构型，其任何有限着色都有单色副本，但存在高密度子集不含该副本，与密度概念互补。
The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances). Answering a question of Erdős and Simonovits [Ars Combin. 9 (1980) 229--246], we show that any graph with less than $\binom{d+2}{2}$ edges has dimension at most $d$. Improving their result, we prove that that the dimension of a graph with maximum degree $d$ is at most $d$. We show the following Ramsey result: if each edge of the complete graph on $2d$ vertices is c…
A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly $1$. A {\em (unit) distance graph} in $\mathbb{R}^d$ is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed $d$ the number of faithful distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1+o(1)) d n \log_2 n}$, and give a shor…
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