Related papers: The Ramsey number of loose cycles versus cliques

Gy\'{a}rf\'{a}s 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…

For integers m >= 1, s >= 0, and t >= 1, let K_s + mK_t denote the join of a clique K_s and m vertex-disjoint copies of K_t. We prove that for fixed m >= 1, t >= 1, and s >= 0, R(K_s + mK_t, K_n) = O( n^{s+t-1} / (log n)^{s+t-2} ). This…

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 $\mathcal{C}^k_n$,…

Gy\'arf\'as, S\'ark\"ozy and Szemer\'edi proved that the $2$-color Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ of a $k$-uniform loose cycle $\mathcal{C}^k_n$ is asymptotically $\frac{1}{2}(2k-1)n,$ generating the same result for…

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…

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…

Let $C_{n}$ be a cycle of length $n$. As an application of Szemer\'{e}di's regularity lemma, {\L}uczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established that…

Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A. %Ruci\'{n}ski, 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…

Burr, Erd\H{o}s, 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…

Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove…

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…

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree,…

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…

A $k$-uniform tight cycle is a $k$-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of $k$ consecutive vertices in that ordering. We show that, for each $k \geq 3$, the Ramsey number of the $k$-uniform…

The wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is…

This paper is a contribution to the investigation of closed partition relations for pairs of countable ordinals. As our main result, we prove that \[\omega^4 \cdot (n-2)+1 < R^{cl}(\omega \cdot n+1,3)<\omega^5\] for every integer $n \geq…

For any positive integers $k$ and $n$, let $B_n^{(k)}$ be the book graph consisting of $n$ copies of the complete graph $K_{k+1}$ sharing a common $K_k$. Let $C_m$ be a cycle of length $m$. Prior work by Allen, \L uczak, Polcyn, and Zhang…

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other:…

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…

For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$…