Related papers: Ramsey numbers for trees II

For $n\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,$ $v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},$…

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2},…

For a graph $H$ and an integer $k\ge1$, let $r(H;k)$ and $r_\ell(H;k)$ denote the $k$-color Ramsey number and list Ramsey number of $H$, respectively. Alon, Buci\'c, Kalvari, Kuperwasser and Szab\'o in 2021 initiated the systematic study of…

The Ramsey numbers $R(T_n,W_8)$ are determined for each tree graph $T_n$ of order $n\geq 7$ and maximum degree $\Delta(T_n)$ equal to either $n-4$ or $n-5$. These numbers indicate strong support for the conjecture, due to Chen, Zhang and…

Let $G_1$ and $G_2$ be two given graphs. 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 $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula…

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…

Let $B_k$ denote a book on $k+2$ vertices and $tB_k$ be $t$ vertex-disjoint $B_k$'s. Let $G$ be a connected graph with $n$ vertices and at most $n(1+\epsilon)$ edges, where $\epsilon$ is a constant depending on $k$ and $t$. In this paper,…

Let $H, H_{1}$ and $H_{2}$ be graphs, and let $H\rightarrow (H_{1}, H_{2})$ denote that any red-blue coloring of $E(H)$ yields a red copy of $H_{1}$ or a blue copy of $H_{2}$. The Ramsey number for $H_{1}$ versus $H_{2}$, $r(H_{1}, H_{2})$,…

Let $n\geq\nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes $\tau_1\geq\tau_2$. Using four natural constructions, we show that the Ramsey number…

For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$.…

Let $G$ and $G_1, G_2, \ldots , G_t$ be given graphs. By $G\rightarrow (G_1, G_2, \ldots , G_t)$ we mean if the edges of $G$ are arbitrarily colored by $t$ colors, then for some $i$, $1\leq i\leq t$, the spanning subgraph of $G$ whose edges…

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the…

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 $(\gamma-1)(\kappa-1)+1$ for arbitrary connected graphs $G_1,\dots,G_n$ and $n_1,\dots,n_r\in\mathbb{N}$, where…

We show that there exists a constant $c>0$ such that every $n$-vertex tree $T$ with $\Delta(T)\le cn$ has Ramsey number $R(T)=\max\{t_1+2t_2,2t_1\}-1$, where $t_1\ge t_2$ are the sizes of the bipartition classes of $T$. This improves an…

The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus the wheel is not yet explored due to the bit more structural…

For two graphs $G$ and $H$, let $r(G,H)$ and $r_*(G,H)$ denote the Ramsey number and star-critical Ramsey number of $G$ versus $H$, respectively. In 1996, Li and Rousseau proved that $r(K_{m},F_{t,n})=tn(m-1)+1$ for $m\geq 3$ and…

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…

In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform…

For given simple graphs $G_1, G_2, \ldots , G_t$, the Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of the complete graph $K_n$ are partitioned into $t$ disjoint color classes giving…

Given graphs $G$, $H_1$, and $H_2$, let $G\xrightarrow{\text{mr}}(H_1,H_2)$ denote the property that in every edge colouring of $G$ there is a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. The constrained Ramsey number, defined as…