Noga Alon, Sonny Ben-Shimon, Michael Krivelevich
We prove that there is an absolute constant $C>0$ so that for every natural $n$ there exists a triangle-free \emph{regular} graph with no independent set of size at least $C\sqrt{n\log n}$.
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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.
A note on the induced Ramsey theorem for spaces
该数学注记针对向量空间的诱导拉姆齐定理，给出了由H. J. Prömel首次证明的该定理的简化证明。
A note on lower bounds for hypergraph Ramsey numbers
本文改进了三色超图Ramsey数的下界，证明了在3-一致的情况下，\[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\]. Erdens和Hajnal的旧界是\[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
A note on Ramsey Numbers for Books
大小为N的书是共享一条公共边的N个三角形的并集。我们证明了，对于所有N>（10^6）M，一本大小为N的书与一本大小为M的书的拉姆齐数等于2N+3。我们的证明是基于计数的。
The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $Γ_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}} - \left( 1/4 - o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gyárfás.
For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by Bohman and Keevash from the random $C_{\ell}$-free process by polylogarithmic factors for all odd $\ell \geq 5$ and $\ell \in \{6,10\}$. For $\ell = 4$ it matches their lower bound from the $C_4$-free process. We also prove, via a different approach, that $r(C_5, t)> (1+o(1))t^{11/8}$ and $r(C_7, t)> (1+o(1))t^{11/9}$. These improve the exponent of $t$ in the p…
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。
For two graphs, $G$, and $H$, an edge-coloring of a complete graph is $(G,H)$-good if there is no monochromatic subgraph isomorphic to $G$ and no rainbow subgraph isomorphic to $H$ in this coloring. The set of number of colors used by some $(G,H)$-colorings of $K_n$ is called a mixed-Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answer is "yes" if $G$ is not a star and $H$ does not contain a pendent edge.
The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = Ω(r^2 n)$.
A note on the logarithmic exponent of Ramsey number $r(4,t)$
该注记结合非对角拉姆齐数下界与Ajtai等人的独立性猜想证明，得出$r(4,t)$的对数指数相关结果，仅作非正式传阅不投稿。
A finite set $A \subset \mathbb{R}^d$ is called $\textit{diameter-Ramsey}$ if for every $r \in \mathbb N$, there exists some $n \in \mathbb N$ and a finite set $B \subset \mathbb{R}^n$ with $\mathrm{diam}(A)=\mathrm{diam}(B)$ such that whenever $B$ is coloured with $r$ colours, there is a monochromatic set $A' \subset B$ which is congruent to $A$. We prove that sets of diameter $1$ with circumradius larger than $1/\sqrt{2}$ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher a…
The restricted online Ramsey numbers were introduced by Conlon, Fox, Grinshpun and He in 2019. In a recent paper, Briggs and Cox studied the restricted online Ramsey numbers of matchings and determined a general upper bound for them. They proved that for $n=3r-1=R_2(r K_2)$ we have $\tilde{R}_{2}(r K_2;n) \leq n-1$ and asked whether this was tight. In this short note, we provide a general lower bound for these Ramsey numbers. As a corollary, we answer this question of Briggs and Cox, and confirm that for $n=3r-1$ we have $\tilde{R}_{2}(r K_2;n) = n-1$. We also show that for $n'=4r-2=R_3(r K_2)…
We say that a graph $F$ strongly arrows a pair of graphs $(G,H)$ if any 2-colouring of its edges with red and blue leads to either a red $G$ or a blue $H$ appearing as induced subgraphs of $F$. The induced Ramsey number, $IR(G,H)$ is defined as the minimum number of vertices of a graph $F$ which strongly arrows a pair $(G,H)$. We will consider two aspects of induced Ramsey numbers. Firstly there will be shown that the lower bound of the induced Ramsey number for a connected graph $G$ with independence number $α$ and a graph $H$ with clique number $ω$ roughly $\frac{ω^2α}{2}$. This bounds is sh…
A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $W_{2n} $ denote an even wheel on $2n+1\ge5$ vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and completely determine the exact value of $GR_k(W_4)$ for all $k\ge2$.
We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for $k$-uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.
This technical note aims at evaluating an asymptotic lower bound on abelian Ramsey lengths.
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。
The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$-uniform hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$ contains an induced monochromatic copy of $F$. We study this function, showing that $r_{\mathrm{ind}}(F)$ is bounded above by a reasonable power of $r(F)$. In particular, our result implies that $r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}$ for any $3$-uniform hypergraph $F$ with $t$ vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on a…
For two graphs $G_1$ and $G_2$ the Ramsey number $R(G_1,G_2)$ is the smallest integer $N$, such that for any graph on $N$ vertices either $G$ contains $G_1$ or $\overline{G}$ contains $G_2$. Let $S_n$ be a star of order $n$ and $W_m$ be a wheel of order $m+1$. In this paper, it is shown that $R(W_n,S_n)\leq{5n/2-1}$, where $n\geq{6}$ is even. It was proven a theorem which implies that $R(W_n,S_n)\geq{5n/2-2}$, where $n\geq{6}$ is even. Therefore we conclude that $R(W_n,S_n)=5n/2-2$ or $5n/2-1$, for $n\geq{6}$ and even.
A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors appear on edges between the parts and at most one color appears on edges in between each pair of parts. We extend this notion by defining a coloring of a complete graph to be $k$-Gallai if every induced subgraph has a nontrivial partition of the vertices such that there are at most $k$ colors present in between parts of the partition. The generalized $(k, \e…
The regular pentagon is canonically Ramsey
本文证明素边数的正多边形是典范拉姆齐集，该类是此前已知典范拉姆齐集类之外的首批新构型，丰富了典范拉姆齐集的研究成果。
In this paper we study a new ideal $\mathcal{WR}$. The main result is the following: an ideal is not weakly Ramsey if and only if it is above $\mathcal{WR}$ in the Katětov order. Weak Ramseyness was introduced by Laflamme in order to characterize winning strategies in a certain game. We apply result of Natkaniec and Szuca to conclude that $\mathcal{WR}$ is critical for ideal convergence of sequences of quasi-continuous functions. We study further combinatorial properties of $\mathcal{WR}$ and weak Ramseyness. Answering a question of Filipów et al. we show that $\mathcal{WR}$ is not $2$-Ramsey,…
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。
A note on zero-sum Ramsey numbers of complete graphs
研究完全图情形下模3的零和拉姆齐数，通过证明对于满足特定条件的n，R(Kn, Z₃)=n + 3，解决了Caro和Mifsud的相关问题。
By a result of Zucker, every Fraïssé structure $\bf F$ for which the elements of $\mathrm{Age}(\bf F)$ have finite Ramsey degrees admits a Fraïssé precompact expansion $\bf F^{*}$ whose age $\mathrm{Age}(\bf F^{*})$ has the Ramsey property. While the original method uses dynamics in spaces of ultrafilters, the purpose of the present short note is to provide a different proof, based on classical tools from Fraïssé theory.
In this note we consider a Ramsey property of random $d$-regular graphs, $\mathcal{G}(n,d)$. Let $r\ge 2$ be fixed. Then w.h.p. the edges of $\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the other hand, there exists a constant $γ> 0$ such that w.h.p., every $r$-coloring of the edges of $\mathcal{G}(n, 2r+1)$ must contain a monochromatic cycle of length at least $γn$. We prove an analogous result for random $k$-out graphs.
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。
A note on tree-cycle Ramsey numbers
本文围绕树 - 圈拉姆齐数展开，Burr等人曾询问满足特定条件的最小函数\(f(m)\)，前人不断改进其上限，本文用不同方法证明\(f(m)\leq2m - 4\)，并提出\(f(m)=\lceil (2m - 1)/3\rceil\)的猜想。
A note on multicolour Ramsey numbers and random sphere graphs
本文通过将二项随机图替换为随机球图，改进了多色Ramsey数的下界，实现了指数级的提升。
Let $r\geq3$ be an integer such that $r-2$ is a prime power and let $H$ be a connected graph on $n$ vertices with average degree at least $d$ and $α(H)\leqβn$, where $0<β<1$ is a constant. We prove that the size Ramsey number \[ \hat{R}({H};r) > \frac{nd}{2}{(r - 2)^2} - C\sqrt n \] for all sufficiently large $n$, where $C$ is a constant depending only on $r$ and $d$. In particular, for integers $k\ge1$, and $r\ge3$ such that $r-2$ is a prime power, we have that there exists a constant $C$ depending only on $r$ and $d$ such that $\hat{R}(P_{n}^{k}; r)> kn{(r - 2)^2}-C\sqrt n -\frac{{({k^2} + k…
A pyramid with a Ramsey base is Ramsey
本文针对欧氏拉姆齐理论中Ivan等人提出的问题，证明了在拉姆齐集的仿射包外添加点得到的棱锥仍为拉姆齐集，解决了拉姆齐集分类的相关问题。
Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$ or $C_7$, respectively. We show that \begin{equation*} R_{k}(C_5;K_m)=Ω(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}), \end{equation*} and \begin{equation*} R_{k}(C_7;K_m)=Ω(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}), \end{equation*} for fixed positive integer $k$ and $m\rightarro…
Some Ramsey results for the n-cube
本文对$n$维立方体的某些子集建立了Ramsey型结果。然后，这可以应用于获得各种相关结构的合理界限，例如大小为3和4的字母表的（部分）Hales-Jewett线，具有小和集的实数集合中的希尔伯特立方体，平面中整数格的“角”，以及整数中的3项等比数列。
Ramsey numbers for regular induced subgraphs
本文研究正则诱导子图的最小顶点数问题，给出k≤5的精确值及k=6,7的下界，并改进了Alon等人的一般下界。
For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G, let the Ramsey number R_r(B(G), k) be the smallest integer n such that no matter how the edges of a complete r-uniform n-vertex hypergraph are colored with k colors, there is a copy of a monochromatic Berge-G subhypergraph. Furthermore, let R(B(G),k) be the smallest integer n such that no matter how all subsets an n-element set are colored with k colors, th…
Ramsey-type problem for an almost monochromatic K_4
本文证明了存在一个常数$c$，使得n > 2的完全图K_n的每一个k-边染色都包含一个边至多接收两种颜色的K_4。这改进了Kostochka和Mubayi的结果，并且是这个问题的第一个指数界。
Infinitely many minimally non-Ramsey size-linear graphs
本文证明了存在无穷多个极小非Ramsey尺寸线性图，肯定回答了Erdős等人提出的相关问题。
In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over $\mathbb{R}$ and ${\mathbb{C}}$, which can be considered metric versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.
For two given graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\bar{G}$ contains a $G_2$. In this note, we determined the Ramsey number $R(K_{1,n},W_m)$ for even $m$ with $n+2\leq m\leq 2n-2$, where $W_m$ is the wheel on $m+1$ vertices, i.e., the graph obtained from a cycle $C_m$ by adding a vertex $v$ adjacent to all vertices of the $C_m$.
The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erdős and Szekeres in 1935, asserts that the Ramsey number of the complete graph with $m$ edges is at most $2^{O(\sqrt{m})}$. Motivated by this estimate Erdős conjectured, more than a quarter century ago, that there is an absolute constant $c$ such that $r(G) \leq 2^{c\sqrt{…
Ramsey Goodness and Beyond
在1983年的一篇开创性论文中，Burr和Erdos开始系统地研究团与大型稀疏图的Ramsey数，提出了一些问题。在本文中，我们开发了一种新的方法，这样的拉姆齐问题使用混合的Szemeredi正则引理，嵌入稀疏图，图兰型稳定性，和其他结构的结果。我们给确切的拉姆齐数的各类图，解决所有，但一个Burr-Erdos问题。
For graphs $G$, $F$ and $H$, let $G\rightarrow (F,H)$ signify that any edge coloring of $G$ in red and blue contains a red $F$ or a blue $H$. The Ramsey number $R(F,H)=\min\{r|\; K_r\rightarrow (F,H)\}$. In this note, we consider redundant edges in Ramsey graphs, which are associate with critical Ramsey numbers. For an integer $k\ge 1$, let ${\mathbb G}=\{G_k,G_{k+1},\dots \}$ be a class of graphs with $δ(G_n)\ge 1$. We define the critical Ramsey number $R_{\mathbb G}(F,H)$ with respect to $\mathbb G$ to be $\max\{n|\; K_r\setminus G_n \rightarrow(F,H),\,G_n\in{\mathbb G}\big\}$, where $r=R(F,…
A note on multidimensional Ramsey numbers
该研究针对多维拉姆齐数问题，证明当$N > 2^{2^{c n^{d-1}}}$时$r$色边的$K_N^d$中存在各方向边单色的$K_n^d$拷贝，改进了此前的指数界与多维Erdős–Szekeres定理的已知界。
Simple proofs of estimations of Ramsey numbers and of discrepancy
本文通过不使用概率语言的简单方法，给出了Ramsey数下界和偏差估计的易懂证明，适合学生等非专业人员理解。
An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $\mathcal{G}$. We show that for every integer $d \geq 3$, almost every $d$-regular graph $G$ satisfies $\overline{R}(\mathcal{G}) \geq \frac{n^{3/2-1/d}}{4\log{n}\log{\log{n}}}$ for every ordering $\mathcal{G}$ of $G$. In particular, there are 3-regular graphs $G$ on $n$ vertic…
For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there is an absolute constant c such that R(G) = R(G,G) < 2c p m for any graph G with m edges and no isolated vertices. Recently this conjecture was proved by B. Sudakov. In this note, using the Sudakovs ideas we give an extension of his result and some interesting corollaries.
The size Ramsey number $\hat{r}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In 1983, Beck provided a beautiful argument that shows that $\hat{r}(P_n)$ is linear, solving a problem of Erdős. In this short note, we provide an alternative but elementary proof of this fact that actually gives a better bound, namely, $\hat{r}(P_n) < 137n$ for $n$ sufficiently large.
Induced Ramsey-type theorems
给出了一种统一的方法来证明具有禁止诱导子图的图的Ramsey型定理，该方法可以用来推广和改进Rodl，Erdos-Hajnal，Promel-Rodl，Nikiforov，Chung-Graham和Luczak-Rodl的早期结果.证明是基于一个简单的引理（推广一个由格雷厄姆，Rodl和Rucinski），可以作为一个替代Szemeredi的正则性引理，从而给更好的界限。同样的方法也可以用来证明伪随机图具有强诱导Ramsey性质。这导致明确的建设上界各种诱导拉姆齐数。
We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$ vertices. The second is the theorem of Cockayne and Lorimer stating that for positive integers satisfying $n_1=\max\{n_1,n_2,\dots,n_t\}$ and with $n=n_1+1+\sum_{i=1}^t (n_i-1)$, the following holds. In every coloring of the edges of $K_n$ with colors $1,2\dots,t$ there is a monochromatic matching of size $n_i$ for some $i\in \{1,2,\dots,t\}$.
In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps in the synthesis of a $2$-colour Ramsey graph $G_1\longrightarrow H$. (The corresponding notion of Ramsey-type numbers was introduced by Erdös, Hajnal and Rado in 1965 and subsequently studied by Erdös and Szemerédi in 1972). For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramsey-infiniteness. From the structural point of view, …
Ramsey numbers and the size of graphs
对于两个图H和G，Ramsey数r（H，G）是使完全图K_n的n个顶点上的每一个红-蓝边染色都包含H的红拷贝或G的蓝拷贝的最小正整数n.受Erdos和Harary问题的启发，本文研究了图G的Ramsey数r（K_s，G）如何依赖于图G的大小.对于s \geq 3，我们证明了：对于任意m条边的G，r（K_s，G）\geq c（m/ \log m）^{ \frac{s+1}{s+3} }对某个只依赖于s的正常数c.这个下界改进了Erdos、Faudree、Rousseau和Schelp的一个较早的结果，并且当s=3时紧到一个多对数因子。我们还研究了r（K_s，G）作为m的函数的最大值。
Ramsey numbers of sparse hypergraphs
本文给出了一个简短的证明：对于适当的常数c（Δ，k），任意n阶k-一致有界度Δ超图H的Ramsey数至多为c（Δ，k）n.这个结果最近被一些作者证明，但这些证明都是基于超图正则性方法的应用。在这里，我们给出了一个更简单，自包含的证明，使用新的技术开发的作者最近与一个参数的Kostochka和Rödl。此外，我们的方法表明，对于k \geq 4，c（Δ，k）\leq 2^{2^{ \Ddots^{2^{c Δ}} }}，其中塔的高度为k，常数c取决于k。它显着改善阿克曼型上界，所产生的正则性证明，我们提出了一个建设表明，至少在某些情况下，这个界是不远处最好的可能。我们的方法也使我们能够证明相当尖锐的结果，最多m边的超图的拉姆齐数。
Linear Ramsey numbers for bounded-degree hypergraphs
我们证明了对于每一个有界度的一致超图，Ramsey数都是线性的。这是一个超图延伸的著名定理为普通图，其中Chvátal等人。显示在1983年。我们的证明是简单的，包含的情况下，并提供了一个强嵌入引理。这表明了作者提出的一个新的超图正则引理的潜力。
A note on the multicolor size-Ramsey numbers of connected graphs
本文研究连通图的多色大小Ramsey数，证明非星形连通图的该数为Ω(r²m)，并加强了树的结论，指出非星形树的Ramsey数与β(T)相关。
Multipartite Ramsey numbers of complete bipartite graphs arising from algebraic combinatorial structures
针对已有方法受限于Hadamard矩阵阶数条件导致的多部拉姆齐数研究遗留问题，该研究引入猜想任意阶存在的$[α]$-Hadamard矩阵推广前人结果，确定了特定完全二部图的两类多部拉姆齐数。
Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$ vertices from which the edges of $K\_{1,s}$ are dropped. In this note we present exact values for $r(K\_m-K\_{1,1},K\_n-K\_{1,s})$ and new upper bounds for $r(K\_m,K\_n-K\_{1,s})$ in numerous cases. We also present some results for the Ramsey number of Wheels versus $K\_n-K\_{1,s}$.
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。
The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from $X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and show that the resulting graph witnesses that $b(2,t) =Ω\left(t^{3/2}/\log t \right)$, thereby improving the best known …
The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a graph $F$ with $m$ edges and no isolated vertices of the form $k^{6km^{2/3}}$ addressing a question of Sudakov [ Adv. Math. 227 (2011), no. 1, 601--609]. Furthermore, the constant in the exponent in the case of bipartite $F$ and two colors is lowered so that $r_2(F)\le 2^{(1+o(1))2\sqrt{2m}}$ improving the result of Alon, Krivelevich and Sudakov [Combin. Pr…
For graphs G and H, let the induced Ramsey number IR(H,G) be the smallest number of vertices in a graph F such that any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G. In this note we consider the case when G=Sn is a star on n edges, for large n, and H is a fixed graph. We prove that (r-1)n < IR(H, Sn) < (r-1)(r-1)n + cn, for any c>0, sufficiently large n, and r denoting the chromatic number of H. The lower bound is asymptotically tight for any fixed bipartite H. The upper bound is attained up to a constant factor, for example by…
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