In this paper, we prove that the multicolored Ramsey number $R(G_1,\dots,G_n,K_{n_1},\dots,K_{n_r})$ is at least $(γ-1)(κ-1)+1$ for arbitrary connected graphs $G_1,\dots,G_n$ and $n_1,\dots,n_r\in\mathbb{N}$, where $γ=R(G_1,\dots,G_n)$ and $κ=R(K_{n_1},\dots,K_{n_r})$. Erd\H os at al. conjectured that $R(C_n,K_l)=(n-1)(l-1)+1$ for every $n\geq l\geq 3$ except for $n=l=3$. Nikiforov proved this conjecture for $n\geq 4l+2$. Using the above bound, we derive the following generalization of this result. $R(C_n,K_{n_1},\dots,K_{n_r})=(n-1)(κ-1)+1$, where $κ=R(K_{n_1},\dots,K_{n_r})$ and $n\geq 4κ+2$.
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.
We find the Ramsey number of a cycle vs. a complete graph when the order of the cycle is at least 4 times as large as the order of the complete graph. This partially confirms a conjecture of Erdős, Faudree, Rousseau, and Schelp made in 1978.
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 $k$-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.
We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp who also proved that cycles (except for $C_4$) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle $C_n$, unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph H is obtained by adding a linear number of chords to a cycle $…
The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtai…
Let $r_k(C_{2m+1})$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge2$, \[r_k(C_{2m+1})<c^{k}\sqrt{k!}\] for all sufficiently large $k$, where $c=c(m)>0$ is a constant. This improves an old result by Bondy and Erdős (Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973) 46-54).
The anti-Ramsey number $Ar(G,H)$ is the maximum number of colors in an edge-coloring of $G$ with no rainbow copy of $H$. In this paper, we determine the exact anti-Ramsey number in the generalized Petersen graph $P_{n,k}$ for cycles $C_d$, where $1\leq k\leq \lfloor \frac{n-1}{2} \rfloor$ and $5\le d \le 6$. We also give an algorithm to obtain the upper bound or lower bound of anti-Ramsey number.
We find the exact value of the Ramsey number $R(C_{2\ell},K_{1,n})$, when $\ell$ and $n=O(\ell^{10/9})$ are large. Our result is closely related to the behaviour of Turán number $ex(N, C_{2\ell})$ for an even cycle whose length grows quickly with $N$.
For two given graphs $G$ and $H$ the planar Ramsey number $PR(G,H)$ is the smallest integer $n$ such that every planar graph $F$ on $n$ vertices either contains a copy of $G$, or its complement contains a copy of $H$. In this paper, we first characterize some structural properties of $C_4$-free planar graphs, and then we determine all planar Ramsey numbers $PR(C_4, W_n)$, for $n\ge 3$.
The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which …
We show that for any positive integer $r$ there exists an integer $k$ and a $k$-colouring of the edges of $K_{2^{k}+1}$ with no monochromatic odd cycle of length less than $r$. This makes progress on a problem of Erdős and Graham and answers a question of Chung. We use these colourings to give new lower bounds on the $k$-colour Ramsey number of the odd cycle and prove that, for all odd $r$ and all $k$ sufficiently large, there exists a constant $ε= ε(r) > 0$ such that $R_{k}(C_{r}) > (r-1)(2+ε)^{k-1}$.
Given graphs $G$ and $H$ and a positive integer $k$, the Gallai-Ramsey number $gr_{k}(G : H)$ is the minimum integer $N$ such that for any integer $n \geq N$, every $k$-edge-coloring of $K_{n}$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. These numbers have recently been studied for the case when $G = K_{3}$, where still only a few precise numbers are known for all $k$. In this paper, we extend the known precise Gallai-Ramsey numbers to include $H = C_{8}$ for all $k$.
Let $H\xrightarrow{s} G$ denote that any $s$-coloring of $E(H)$ contains a monochromatic $G$. The degree Ramsey number of a graph $G$, denoted by $R_Δ(G, s)$, is $\min \{Δ(H): H \xrightarrow{s} G \}$. We consider degree Ramsey numbers where $G$ is a fixed even cycle. Kinnersley, Milans, and West showed that $R_Δ(C_{2k},s) \geq 2s$, and Kang and Perarnau showed that $R_Δ(C_4, s) = Θ(s^2)$. Our main result is that $R_Δ(C_6, s) = Θ(s^{3/2})$ and $R_Δ(C_{10}, s) = Θ(s^{5/4})$. Additionally, we substantially improve the lower bound for $R_Δ(C_{2k}, s)$ for general $k$.
We determine several generalised Ramsey numbers for two sets $Γ_1$ and $Γ_2$ of cycles, in particular, all generalised Ramsey numbers $R(Γ_1,Γ_2)$ such that $Γ_1$ or $Γ_2$ contains a cycle of length at most $6$, or the shortest cycle in each set is even. This generalises previous results of Erdős, Faudree, Rosta, Rousseau, and Schelp from the 1970s. Notably, including both $C_3$ and $C_4$ in one of the sets, makes very little difference from including only $C_4$. Furthermore, we give a conjecture for the general case. We also describe many $(Γ_1,Γ_2)$-avoiding graphs, including a complete char…
Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. Rödl, A. %Ruciński, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of 3-uniform loose cycles on $2n$ vertices is asymptotically $\frac{5n}{2}$. Their proof is based on the method of Regularity Lemma. Here, without using this method, we generalize their result by determining the exact values of 2-color Ramsey numbers involving loose paths and cycles in 3-uniform hypergraphs. More precisely, we prove that for every $n\geq m\geq 3$, $R(\mathc…
Given two graphs $G$ and $H$ and a positive integer $k$, the $k$-color Gallai-Ramsey number, denoted by $gr_{k}(G : H)$, is the minimum integer $N$ such that for all $n \geq N$, every $k$-coloring of the edges of $K_{n}$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. We prove that $gr_{k} (K_{3} : C_{2\ell + 1}) = \ell \cdot 2^{k} + 1$ for all $k \geq 1$ and $\ell \geq 3$.
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)\leq R_k(P_n)\leq R_k(C_n)\leq kn+o(n)$. The upper bound was recently improved by Sárközy who showed that $R_k(C_n)\leq\left(k-\frac{k}{16k^3+1}\right)n+o(n)$. Here we show $R_k(C_n) \leq (k-\frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G, H)$ is defined as $r(G, H) = min\{|V (F)|: F \rightarrow (G, H)\}$, while the restricted size Ramsey number $r^{*}(G, H)$ is defined as $r^{*}(G, H) = min\{|E (F)|: F \rightarrow (G, H) , |V (F) | = r(G, H)\}$. In this paper we determine previously unknown restricted size Ramsey numbers $r^{*}(P_3, C_n)$ for $7 \leq n \leq 12$. We also give new upper bound $r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$.
Ramsey number of a cycle versus a graph of a given size
KU Leuven Campus Kulak-Kortrijk（荷语鲁汶大学库拉-科特赖克校区）；HUN-REN, Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络，阿尔弗雷德·雷尼数学研究所）；ETH Zürich（苏黎世联邦理工学院）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；University College London（伦敦大学学院）
For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting of $n$ triangles all sharing a common vertex. In this paper, we prove that for all sufficiently large $n$, \[ R(C_{2\lfloor an\rfloor}, F_n)= \left\{ \begin{array}{ll} (2+2a+o(1))n & \textrm{if $1/2\leq a< 1$,}\\ (4a+o(1))n & \textrm{if $ a\geq 1$.} \end{array} \right. \]
The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study …
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…
In this paper, we consider an extension of cycle-complete graph Ramsey numbers to Berge cycles in hypergraphs: for $k \geq 2$, a {\em non-trivial Berge $k$-cycle} is a family of sets $e_1,e_2,\dots,e_k$ such that $e_1 \cap e_2, e_2 \cap e_3,\dots,e_k \cap e_1$ has a system of distinct representatives and $e_1 \cap e_2 \cap \dots \cap e_k = \emptyset$. In the case that all the sets $e_i$ have size three, let $\mathcal{B}_k$ denotes the family of all non-trivial Berge $k$-cycles. The {\em Ramsey numbers} $R(t,\mathcal{B}_k)$ denote the minimum $n$ such that every $n$-vertex $3$-uniform hypergrap…
For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\ge4$ vertices and let $Θ_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. Unlike Ramsey number of odd cycles, little is known about the general behavior of $R_k(C_{2n})$ except that $R_k(C_{2n})\ge (n-1)k+n+k-1$ for all $k\ge2$ and $n\ge2$. In this paper, we study Ramsey number of even c…
For graphs $H_1,H_2$ by $r^*(H_1,H_2)$ we denote the minimum number of edges in a graph $G$ on $r(H_1,H_2)$ vertices such that $G\to (H_1,H_2)$. We show that for each pair of natural numbers $k,n$, $k\le n$, where $k$ is odd and $n$ is large enough, we have $$r^*(C_n,C_k)=\lceil (n+1)(2n-1)/2\rceil \,.$$
A $k$-uniform loose cycle $\mathcal{C}_n^k$ is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ edges $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots,v_{(i-1)(k-1)+k}\}$, $1\leq i\leq n$, where we use mod $n(k-1)$ arithmetic. The Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ is asymptotically $\frac{1}{2}(2k-1)n$ as has been proved by Gyárfás, Sárközy and Szemerédi. In this paper, we investigate to determining the exact value of diagonal Ramsey number of $\mathcal{C}^k_n$ and we show that for $n\geq 2$ and $k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=…
We study the structure of red-blue edge colorings of complete graphs, with no copies of the $n$-cycle $C_n$ in red, and no copies of the $n$-wheel $W_n = C_n \ast K_1$ in blue, for an odd integer $n$. Our first main result is that in any such coloring, deleting at most two vertices we obtain a vertex-partition of $G$ into three sets such that the edges inside the partition classes are red, and edges between partition classes are blue. As a second result, we obtain bounds for the Ramsey numbers of $r(C_{2k+1},W_{2j})$ for $k < j$ integers, which asymptotically confirm the values of $4j+1$, as i…
Let $G$, $H$ and $K$ represent three graphs without loops or parallel edges and $n$ represent an integer. Given any red blue coloring of the edges of $G$, we say that $K \rightarrow (G,H)$, if there exists red copy of $G$ in $K$ or a blue copy of $H$ in $K$. Let $K_n$ represent a complete graph on $n$ vertices, $C_n$ a cycle on $n$ vertices and $S_n=K_{1,n}$ a star on $n+1$ vertices. The Ramsey number $r(G, H)$ is defined as $\min\{n \mid K_n\rightarrow (G,H)\}$. Likewise, the star-critical Ramsey number $r_*(H, G)$ is defined $\min\{k \mid K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (H, G) \}$. W…
If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{\vec H}=p_{\vec H}(n)$ for the property $G(n,p)\to \vec H$ for the cases in which $\vec H$ is an acyclic orientation of a complete graph or of a cycle.
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…
The Multicolour Size--Ramsey Number of an Even Cycle
School of Mathematics, Nanjing University（南京大学数学系）；ECOPRO, Institute for Basic Science（基础科学研究所ECOPRO）；School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）
For given graphs $G_1, G_2,\ldots, G_n$ and any integer $j$, the size of the multipartite Ramsey number $m_j(G_1, G_2,\ldots, G_n)$ is the smallest positive integer $t$ such that any $n$-coloring of the edges of $K_{j\times t}$ contains a monochromatic copy of $G_i$ in color $i$ for some $i$, $1 \leq i \leq n$, where $K_{j\times t}$ denotes the complete multipartite graph having $j$ classes with $t$ vertices per each class. In this paper we compute the size of the multipartite Ramsey number $m_j(K_{1,2}, P_4, nK_2)$ for any $j,n\geq 2$ and $m_j(nK_2,C_7)$, for any $j\leq4$ and $n\geq 2$.
A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given graph $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 $C_m$ denote the cycle on $m\ge4$ vertices and let $Θ_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(Θ_{2n+1})=n\cdot …
Recently, determining the Ramsey numbers of loose paths and cycles in uniform hypergraphs has received considerable attention. It has been shown that the $2$-color Ramsey number of a $k$-uniform loose cycle $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$, is asymptotically $\frac{1}{2}(2k-1)n$. Here we conjecture that for any $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{P}^k_n,\mathcal{P}^k_m)=R(\mathcal{P}^k_n,\mathcal{C}^k_m)=R(\mathcal{C}^k_n,\mathcal{C}^k_m)+1=(k-1)n+\lfloor\frac{m+1}{2}\rfloor.$$ Recently the case $k=3$ is proved by the authors. In this paper, first we show that th…
Gyárfás et al. determined the asymptotic value of the diagonal Ramsey number of $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n),$ generating the same result for $k=3$ due to Haxell et al. Recently, the exact values of the Ramsey numbers of 3-uniform loose paths and cycles are completely determined. These results are motivations to conjecture that for every $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_m)=(k-1)n+\lfloor\frac{m-1}{2}\rfloor,$$ as mentioned by Omidi et al. More recently, it is shown that this conjecture is true for $n=m\geq 2$ and $k\geq 7$ and for $k=4$…
For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large, \[ R_k(C_n)=2^{k-1}(n-1)+1. \] This resolves a conjecture of Bondy and Erdős [J. Combin. Th. Ser. B \textbf{14} (1973), 46--54] for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimis…
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…
For graphs $G$ and $H$, let $G \overset{\mathrm{rb}}{\longrightarrow} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, Škorić and Steger [J. Combin. Theory Ser. B 124 (2017),1-38], we determine the threshold for $G(n,p) \overset{\mathrm{rb}}{\longrightarrow} C_\ell$ for cycles $C_\ell$ of any given length $\ell \geq 4$.
We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by Sárközy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a l…
We determine the $2$-color Ramsey number of a {\em connected} triangle matching $c(nK_3)$ which is any connected graph containing $n$ vertex disjoint triangles. We obtain that $R(c(nK_3),c(nK_3))=7n-2$, somewhat larger than in the classical result of Burr, Erd\H os and Spencer for a triangle matching, $R(nK_3,nK_3)=5n$. The motivation is to determine the Ramsey number $R(C_n^2,C_n^2)$ of the square of a cycle $C_n^2$. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an "almost" square of a cycle $C_n^{2,c}$ (a cycle of length $n$ in which all but at…
Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 50m$. Combining this with a result of Faudree, Lawrenc…
The generalized Ramsey number $R(G_1, G_2)$ is the smallest positive integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1}, W_{n})$ of odd cycles versus larger wheels, leaving open the particular case where $n = 2j$ is even and $k<j<3k/2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1}, W_{2j})=4j+1$. In 2015, Sanhueza-M…
Let $K_n$ denote the complete graph on $n$ vertices and $G, H$ be finite graphs. Consider a two-coloring of edges of $K_n$. When a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue is in $K_n$, we write $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the smallest positive integer $n$ such that $K_{n} \rightarrow (G, H)$. Star critical Ramsey $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)$. In this paper, we find $r_*(C_n, K_m)$ for $m \geq 7$ and $n \geq (m-3)(m-1)$.
The 3-uniform tight cycle $C_s^3$ has vertex set $ Z_s$ and edge set $\{\{i, i+1, i+2\}: i \in Z_s\}$. We prove that for every $s \not\equiv 0$ (mod 3) and $s \ge 16$ or $s \in \{8,11,14\}$ there is a $c_s>0$ such that the 3-uniform hypergraph Ramsey number $$r(C_s^3, K_n^3)< 2^{c_s n \log n}$$ This answers in strong form a question of the author and Rödl who asked for an upper bound of the form $2^{n^{1+ε_s}}$ for each fixed $s \ge 4$, where $ε_s \rightarrow 0$ as $s \rightarrow \infty$ and $n$ is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in …
Online Ramsey game is played between Builder and Painter on an infinite board $K_{\mathbb N}$. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs $H_1$ and $H_2$. Builder aims to create either a red copy of $H_1$ or a blue copy of $H_2$ in $K_{\mathbb N}$ as soon as possible, and Painter tries to prevent it. The online Ramsey number $\tilde{r}(H_1,H_2)$ is the minimum number of rounds such that the Builder wins. We study $\tilde{r}(C_k,C_n)$ where $k$ is fixed and $n$ is large. We show that $\tilde{r}(C_k,C_n)=2n+\mathcal O(k)$ for an absolute c…
For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $ contains a monochromatic copy of $G_i$ of color $i$ for some $1\leq i\leq k$. We denote $\hat{R}(G_1,\ldots,G_k)$ by $\hat{R}_{k}(G)$ when $G_1=\cdots=G_k=G$. Haxell, Kohayakawa and Łuczak showed that the size Ramsey number of a cycle $C_n$ is linear in $n$ i.e. $\hat{R}_{k}(C_{n})\leq c_k n$ for some constant $c_k$. Their proof, is based on the regularity lemma…
For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $Φ:E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq Φ(e)$. We denote by $\mathcal{B}^rG$, the family of $r$-uniform Berge-$G$ hypergraphs. For families $\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t$ of $r$-uniform hypergraphs, the Ramsey number $R(\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t)$ is the smallest integer $n$ such that in every $t$-hyperedge coloring of $\mathcal{K}_{n}^r$ there is a monochromatic copy of a hypergraph in $\mathcal{H}…
The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(χ(H)-1)+σ(H)$, where $σ(H)$ is the minimum possible size of a colour class in a $χ(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|χ(H)$. We improve this 40-year-old result of Bu…
The size-Ramsey number $\hat{R}(\mathcal{F},H)$ of a family of graphs $\mathcal{F}$ and a graph $H$ 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, say, red and blue, yields a red copy of a graph from $\mathcal{F}$ or a blue copy of $H$. In this paper we first focus on $\mathcal{F} = \mathcal{C}_{\le cn}$, where $\mathcal{C}_{\le cn}$ is the family of cycles of length at most $cn$, and $H = P_n$. In particular, we show that $2.00365 n \le \hat{R}(\mathcal{C}_{\le n},P_n) \le 31n$. Using simil…
Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ(H)$ is the size of the smallest color class in a $χ(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(χ(H)-1)+σ(H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively st…
For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at}<r(C^3_s, K^3_t)<2^{bt^2\log t}.$$ The lower bound is obtained via a probabilistic construction. The upper bound for $s>5$ is proved by using supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for $s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we de…
Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey number $\tilde{r}(G,H)$. In this paper, we consider the case where $G$ is a path $P_k$. We prove that $\tilde{r}(P_3, P_{\ell+1}) = \lceil 5\ell/4 \rceil = \tilde{r}(P_3, C_\ell)$ for all $\ell \ge 5$…
