Vladimir Nikiforov, Cecil Rousseau
A book of size N is the union of N triangles sharing a common edge. We show that the Ramsey number of a book of size N vs. a book of size M equals 2N+3 for all N>(10^6)M. Our proof is based on counting.
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A book of size q is the union of q triangles sharing a common edge. We find the exact Ramsey number of books of size q versus books of size p when p<q/6-o(q).
Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for books $B_{m} = K_{2} + \overline{K_{m}}$ and prove sharp results for $m \leq 5$.
The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. The Ramsey numbers of $B_n^{(k)}$ are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed $k$, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi's regularity lemma, thus providing much tighter …
A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies $r(B_m,B_n)\le 2(m+n)+c$ for some constant $c>0$. In this paper, we obtain that $r(B_m, B_n)\le 2(m+n)+o(n)$ for all $m\le n$ and $n$ large, which confirms the conjecture of Rousseau and Sheehan asymptotically. As a corollary, our result implies that a related conjecture of Faudree, Rousseau and Sheehan (1982) on strongly regular graph holds asymptotically.
The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n^{(k)}, B_n^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn}^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being $k$-partite, while, for $c$ bounded away from $0$, random colorings of an appropriate density ar…
We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book $B_n^{(k)}$ to be the graph consisting of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, we show that the Ramsey number $r(B_n^{(k)}) = 2^k n + o_k(n)$. In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic ver…
Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $\frac{8}{9}n+112\le m\le \lceil\frac{3n}{2}\rceil+1$ and $n \geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {\it Ars Combin.,} {\bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k \geq 3$. Name…
Lower Bounds for Book Ramsey Numbers
该研究通过块循环构造和SAT求解器，确定了书图Ramsey数的下界，并枚举了临界图数量。
An r-book of size q is a union of q (r+1)-cliques sharing a common r-clique. We find exactly the Ramsey number of a p-clique versus r-books of sufficiently large size. Furthermore, we find asymptotically the Ramsey number of any fixed p-chromatic graph versus r-books of sufficiently large size. The key element in our proofs is Szemeredi's Regularity Lemma.
A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of $r(C_4,B_n)$ for $1\le n\le 14$. We aim to show the exact value of $r(C_4,B_n)$ for infinitely many $n$. To achieve this, we first prove that $r(C_4,B_{(m-1)^2+(t-2)})\le m^2+t$ for $m\ge4$ and $0 \leq t \leq m-1$. This improves upon a result by Faudree, Rousseau and Sheehan (1978) which states that \begin{align*} r(C_4,B_n)\le g(g(n)), \;\;\text{where}\;\;g(n)=n+\lfloor\sqrt{n-1}\rfloor+2. \end{align*} Com…
Small Ramsey numbers for books, wheels, and generalizations
本文通过多种方法确定了书籍和轮子的拉姆齐数，并研究了广义拉姆齐数的某些新界限。
Ramsey numbers of sparse graphs versus disjoint books
该论文证明了对任意稀疏连通图G，其与t个不相交书图的Ramsey数等于2n+t-2，推广了Erdős等人关于树情形的结果。
Weakened Gallai-Ramsey Numbers for Books
本文研究书图B_n的弱化Gallai-Ramsey数，扩展了前人t=3时的结果，确定了t>3时gr^3_2(B_n)在5≤n≤15的值，并给出该类数的一般下界。
Sharp Ramsey thresholds for large books
本文研究随机图中大书图 $B_n^{(k)}$ 的 Ramsey 性质，确定其边概率尖锐阈值，并推出渐近 Ramsey 数 $r(B_n^{(k)},B_n^{(k)})=2^kn+o(n)$。
Ramsey numbers of long even cycles versus books
本文研究长偶圈与 $k$ 书图之间的 Ramsey 数，在 $m=\Omega(n^{9/10})$ 等渐近条件下，通过显式构造和半随机图分析完整确定了固定 $k\ge3$ 时 $R(C_m,B_n^{(k)})$ 的精确值。
Star-critical Ramsey numbers involving large generalized books
本文针对满足特定条件的完全多部图G与广义书图H，在未使用Szemerédi正则性引理的情况下，推导得到了对应的星临界拉姆齐数表达式。
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。
Book Ramsey numbers via algebraic constructions
本文通过构造新的强正则图族，证明了无穷多个n满足R(B_n,B_n)=4n+1，并改进了R(B_{n-2},B_n)的上界，去除了原有模条件。
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
In 1983, Burr and Erdős initiated the study of Ramsey goodness problems.Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erdős, in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let $B_{k,n}$ be the book graph on $n$ vertices which consists of $n-k$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $H=K_p(a_1,\dots,a_{p})$ be the complete $p$-partite graph with parts of sizes $a_1,\dots,a_{p}$. Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2021) revisit several Ramsey go…
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 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…
The Ramsey number of the 4-cycle versus a book graph
该研究针对k≥3的情况，通过改进无C4图的相关不等式，得到了无穷多n对应的4-环与书图的拉姆齐数r(C4,B_n^(k))的精确值及通用上下界。
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.
The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of $H$ in every two-coloring of $E(G)$. The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex …
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.
New upper bound for multicolor Ramsey numbers
本研究针对多色Ramsey数上界问题，结合可变阶正系数根滤波器与多色书方法的保留脊细化技术，证明了满足一定参数条件的对角r色图Ramsey数新上界公式。
Optimizing the CGMS upper bound on Ramsey numbers
本文简化了Campos等人关于Ramsey数上界的突破性证明，用归纳陈述替代书算法，优化参数后得到对角Ramsey数上界$R(k,k) \leq (3.8)^{k+o(k)}$，还将非对角Ramsey数上界推广到多色情形。
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…
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。
A recent breakthrough of Conlon and Ferber yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers. In this note, we modify their construction and obtain improved bounds for more than three colors.
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,…
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。
Ramsey goodness of books revisited
本文重新研究书图的 Ramsey 良性质，用不依赖 Szemerédi 正则性引理的简短证明改进规模界，并推广到完全多部图情形。
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 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 $C\sqrt{n\log n}$.
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。
On the size edge-ordered Ramsey numbers of graphs
本文通过正则性引理证明了稀疏图家族的边有序Ramsey数具有非线性增长，同时展示了三个特定边有序图家族的Ramsey数具有线性或近线性特性。
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…
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…
Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a monochromatic copy of $P_t$. In this note, we determine $R_r(P_5)$, which is approximately $3r$.
In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, "corner" in the integer lattice in the plane, and 3-term geometric progressions in integers.
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…
In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.
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。
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 $C_m$.
Let $H\xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_Δ(G;s)$ is defined to be $\min\{Δ(H):H\xrightarrow{s} G\}$, and the degree bipartite Ramsey number $br_Δ(G;s)$ is defined to be $\min\{Δ(H):H\xrightarrow{s} G\; \mbox{and} \;χ(H)=2\}$. In this note, we show that $r_Δ(K_{m,n};s)$ is linear on $n$ with $m$ fixed. We also determine $br_Δ(G;s)$ where $G$ are trees, including stars and paths, and complete bipartite graphs.
Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers
本文研究了超图Ramsey数的递归上界，提出改进的递推关系并应用于超图Ramsey数的上界优化。
An update on multicolor Ramsey lower bounds
基于球面随机几何图构造，改进了多色Ramsey数的下界，提供了更精确的理论结果。
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.
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