Book Ramsey numbers I

Large book--cycle Ramsey numbers

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…

书图Ramsey数的下界

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。

Large generalized books are p-good

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.

Ramsey numbers of quadrilateral versus books

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…

书籍、轮子及一般化的小拉姆齐数

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。

稀疏图与不相交书图之间的Ramsey数

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。

书图的弱化Gallai-Ramsey数

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。

大书图的尖锐 Ramsey 阈值

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 $r(B_n^{(k)},B_n^{(k)})=2^kn+o(n)$。

长偶圈与书图的 Ramsey 数

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。

涉及大型广义书图的星临界拉姆齐数

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。

通过代数构造的Book拉姆齐数

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。

Ramsey non-goodness involving books

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…

4-环与书图的拉姆齐数

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。

Three early problems on size Ramsey numbers

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 …

多色Ramsey数的新上界

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。

优化Ramsey数的CGMS上界

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。

重新审视书图的 Ramsey 良性质

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。

关于图的大小边有序Ramsey数

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。

Complementary Ramsey numbers and Ramsey graphs

In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $\bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $\bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.

New Lower Bound Formulas for Multicolored Ramsey Numbers

We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers.

Ramsey and Gallai-Ramsey number for wheels

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} 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 $G$. Much like graph Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being very difficult to compute in general. As yet, still only precious few sharp results are known. In this paper, we obtain bounds on the Gallai-Ramsey number for wheels and the exact value for the wheel on $5$ vertices.

Anti-Ramsey numbers for paths

We determine the anti-Ramsey numbers for paths. This confirms a conjecture posed by Erdős, Simonovits and Sós in 1970s.

Ramsey and Gallai-Ramsey numbers for two classes of unicyclic graphs

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} 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 consider two classes of unicyclic graphs, the star with an extra edge and the path with a triangle at one end. We provide the $2$-color Ramsey numbers for these two classes of graphs and use these to obtain general upper and lower bounds on the Gallai-Ramsey numbers.

Difference Ramsey Numbers and Issai Numbers

We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers. Using notions from this algorithm we then give some results for generalized Schur numbers, which we call Issai numbers.

Lower bounds for multicolor Ramsey numbers

We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.

Ramsey numbers for bipartite graphs with small bandwidth

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.

Ramsey numbers for partially-ordered sets

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.

An improved lower bound for multicolor Ramsey numbers and the half-multiplicity Ramsey number problem

The multicolor Ramsey number problem asks, for each pair of natural numbers $\ell$ and $t$, for the largest $\ell$-coloring of a complete graph with no monochromatic clique of size $t$. Recent works of Conlon-Ferber and Wigderson have improved the longstanding lower bound for this problem. We make a further improvement by replacing an explicit graph appearing in their constructions by a random graph. Graphs useful for this construction are exactly those relevant for a problem of Erdős on graphs with no large cliques and few large independent sets. We also make some basic observations about thi…

Ordered Ramsey numbers of loose paths and matchings

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set $\{1,\ldots,N\}$ contains a monochromatic copy of $G$ whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual graph Ramsey numbers. We determine that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height one less than the maximum degree. We al…

On the size-Ramsey number of hypergraphs

The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for $k$-uniform hypergraphs. Analogous to the graph case, we consider the size-Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size-Ramsey numbers for hypergraphs are extremely difficult to determine, and many…

Two multicolor Ramsey numbers

Two new bounds for multicolor Ramsey numbers are proved: $R(K_3,K_3,C_4,C_4)\geq 27$ and $R_4(C_4)\leq 19$.

Three colour bipartite Ramsey number of cycles and paths

The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $n$ for which every $k$-edge-coloured complete bipartite graph $K_{n,n}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour Ramsey number of paths. In this paper we determine asymptotically the $3$-colour bipartite Ramsey number of paths and (even) cycles.

Multicolour bipartite Ramsey number of paths

The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the $2$-colour bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine a…

Ramsey and Gallai-Ramsey numbers for stars with extra independent edges

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} 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 the graph $G = S_t^{r}$ obtained from a star of order $t$ by adding $r$ extra independent edges between leaves of the star so there are $r$ triangles and $t - 2r - 1$ pendent edges in $S_t^{r}$. We also prove some sharp results when $t = 2$.

Mixed, Multi-color, and Bipartite Ramsey Numbers Involving Trees of Small Diameter

In this paper we study Ramsey numbers for trees of diameter 3 (bistars) vs., respectively, trees of diameter 2 (stars), complete graphs, and many complete graphs. In the case of bistars vs. many complete graphs, we determine this number exactly as a function of the Ramsey number for the complete graphs. We also determine the order of growth of the bipartite $k$-color Ramsey number for a bistar.

递归上界与顶点在线Ramsey游戏的递归上界及其在超图Ramsey数中的应用

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。

Multicolor Ramsey numbers for Berge cycles

In this paper, for small uniformities, we determine the order of magnitude of the multicolor Ramsey numbers for Berge cycles of length $4$, $5$, $6$, $7$, $10$, or $11$. Our result follows from a more general setup which can be applied to other hypergraph Ramsey problems. Using this, we additionally determine the order of magnitude of the multicolor Ramsey number for Berge-$K_{a,b}$ for certain $a$, $b$, and uniformities.

The Ramsey number of dense graphs

The Ramsey number r(H) of a graph H is the smallest number n such that, in any two-colouring of the edges of K_n, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t vertices and density \r, proving that r(H) \leq 2^{c \sqrt{\r} \log (2/\r) t}. We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree \r t and the Ramsey number of random graphs in \mathcal{G}(t, \r), that is, graphs on t vertices where each edge has been chosen independently with probability \r.

List Ramsey numbers

We introduce the list colouring extension of classical Ramsey numbers. We investigate when the two Ramsey numbers are equal, and in general, how far apart they can be from each other. We find graph sequences where the two are equal and where they are far apart. For $\ell$-uniform cliques we prove that the list Ramsey number is bounded by an exponential function, while it is well-known that the Ramsey number is super-exponential for uniformity at least $3$. This is in great contrast to the graph case where we cannot even decide the question of equality for cliques.

The Ramsey number for a triple of large cycles

We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different parity.

Regressive Ramsey numbers are Ackermannian

We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques.

Projective planes and set multipartite Ramsey numbers for $C_4$ versus star

Set multipartite Ramsey numbers were introduced in 2004, ge-neralizing the celebrated Ramsey numbers. Let $C_4$ denote the four cycle and let $K_{1,n}$ denote the star on $n+1$ vertices. In this paper we investigate bounds on $C_4-K_{1,n}$ set multipartite Ramsey numbers. Relationships between these numbers and the classical $C_4-K_{1,n}$ Ramsey numbers are explored. Then several near-optimal or exact classes are derived as applications. As the main goal, polarity graphs from projective planes allow us to find suitable subgraphs which yield some optimal classes too.

On the Ramsey Numbers of Trees with Small Diameter

We estimate the Ramsey number r(T) = r(T,T) for various trees T, obtaining a precise value for r(T) for a large number of trees of diameter 3. Furthermore we prove that all trees of diameter 3 are Ramsey unsaturated as defined by Balister, Lehel, and Schelp in their article "Ramsey unsaturated and saturated graphs."

经典非拉姆齐问题的拉姆齐理论刻画

Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems

本文提出更一般的经典拉姆齐数概念并用代数工具给出初步刻画，同时论证拉姆齐对象的结构可处理性，并说明格林-陶定理、孪生素数猜想、张益唐定理和波利尼亚克猜想均可视为关于拉姆齐数的陈述。

A New Upper Bound for Diagonal Ramsey Numbers

We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C \frac{\log k}{\log \log k}} \binom{2k}{k}.\]

On the Ramsey numbers for a combination of paths and Jahangirs

For given graphs $G$ and $H,$ the \emph{Ramsey number} $R(G,H)$ is the least natural number $n$ such that for every graph $F$ of order $n$ the following condition holds: either $F$ contains $G$ or the complement of $F$ contains $H.$ In this paper, we improve the Surahmat and Tomescu's result \cite{ST:06} on the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number $R(\cup G,H)$, where $G$ is a path and $H$ is a Jahangir graph.

An improved lower bound on multicolor Ramsey numbers

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.

部分超图的典范拉姆齐数

Canonical Ramsey numbers for partite hypergraphs

本文证明对于任意固定一致性的部分超图，其典范拉姆齐数以单指数增长。

Gallai-Ramsey numbers for graphs with five vertices of chromatic number four

Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we determine the Gallai-Ramsey numbers for connected graphs with five vertices of chromatic number four.

The multipartite Ramsey number for the 3-path of length three

We study the Ramsey number for the 3-path of length three and $n$ colors and show that $R(P^3_3;n)\le λ_0 n+7\sqrt{n}$, for some explicit constant $λ_0=1.97466\dots$.

Constructive Methods in Gallai-Ramsey Theory for Hypergraphs

Much recent progress in hypergraph Ramsey theory has focused on constructions that lead to lower bounds for the corresponding Ramsey numbers. In this paper, we consider applications of these results to Gallai colorings. That is, we focus on the Ramsey numbers resulting from only considering $t$-colorings of the hyperedges of complete $r$-uniform hypergraphs in which no rainbow $K_{r+1}^{(r)}$-subhypergraphs exists. We also provide new constructions which imply improved lower bounds for many $3$ and $4$-uniform Ramsey numbers and $3$ and $4$-uniform Gallai-Ramsey numbers.

New Lower Bounds for Some Multicolored Ramsey Numbers

We use finite fields and extend a result of Fan Chung to give eight new, nontrivial, lower bounds.

A Ramsey-type theorem for the matching number regarding connected graphs

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $ρ$, every graph $G$ with sufficiently large $ρ(G)$ contains a `well-structured' induced subgraph $H$ with large $ρ(H)$. The classical Ramsey's theorem deals with the case when the graph parameter under consideration is the number of vertices; there is also a Ramsey-type theorem regarding connected graphs. Given a graph $G$, the matching number and the induced matching number of $G$ is the maximum size of a matching and an induced matching, respectively, of $G$. In thi…

Ordered Ramsey numbers

Given a labeled graph $H$ with vertex set $\{1, 2,\ldots,n\}$, the ordered Ramsey number $r_<(H)$ is the minimum $N$ such that every two-coloring of the edges of the complete graph on $\{1, 2, \ldots,N\}$ contains a copy of $H$ with vertices appearing in the same order as in $H$. The ordered Ramsey number of a labeled graph $H$ is at least the Ramsey number $r(H)$ and the two coincide for complete graphs. However, we prove that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices. Among other results, we also prove a general upper …