相关论文: Some Upper Bounds on Ramsey Numbers Involving $C_4…

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

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The path-critical Ramsey number $R_{\pi}(G,H)$ is the largest $n$ such that any red-blue…

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…

The size Ramsey number $ \hat{r}(G,H) $ of two graphs $ G $ and $ H $ is the smallest integer $ m $ such that there exists a graph $ F $ on $ m $ edges with the property that every red-blue colouring of the edges of $ F $, yields a red copy…

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…

A $(k+r)$-uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$-daisy if there exists a partition of the vertices $V(H)=K\cup M$ with $|K|=k$, $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$-tuples $K\cup P$, where $P$ is…

We present some new constructive upper bounds based on product graphs for generalized vertex Folkman numbers. They lead to new upper bounds for some special cases of generalized edge Folkman numbers, including $F_e(K_3,K_4-e; K_5) \leq 27$…

Let $f(n,p,q)$ denote the minimum number of colors needed to color the edges of $K_n$ so that every copy of $K_p$ receives at least $q$ distinct colors. In this note, we show $\frac{6}{7}(n-1) \leq f(n,5,8) \leq n + o(n)$. The upper bound…

We focus on two hypergraph Ramsey problems. First, we consider the Erd\H{o}s-Hajnal function $r_k(k+1,t;n)$. In 1972, Erd\H{o}s and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\le t\le k$. To finish…

Given two graphs $G_1, G_2$, the connected size Ramsey number ${\hat{r}}_c(G_1,G_2)$ is defined to be the minimum number of edges of a connected graph $G$, such that for any red-blue edge colouring of $G$, there is either a red copy of…

Computer-based attempts to construct lower bounds for small Ramsey numbers are discussed. A systematic review of cyclic Ramsey graphs is attempted. Many known lower bounds are reproduced. Several new bounds are reported.

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 -…

A graph is $(t_1, t_2)$-Ramsey if any red-blue coloring of its edges contains either a red copy of $K_{t_1}$ or a blue copy of $K_{t_2}$. The size Ramsey number is the minimum number of edges contained in a $(t_1,t_2)$-Ramsey graph.…

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest integer $N$ such that every graph $\Gamma$ on $N$ vertices contains $G$ or its complement $\overline{\Gamma}$ contains $H$ as a subgraph. In graph Ramsey theory, the…

For a graph $H$ and an integer $n$, we let $nH$ denote the disjoint union of $n$ copies of $H$. In 1975, Burr, Erd\H{o}s, and Spencer initiated the study of Ramsey numbers for $nH$, one of few instances for which Ramsey numbers are now…

We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$…

The known lower bound for the the classical Ramsey number $R(5,6)$ is improved from $58$ to $59$. The method used to construct the graph is a simple variant of computational methods that have been previously used to construct Ramsey graphs.…

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute…

The areas of Ramsey theory and random graphs have been closely linked ever since Erd\H{o}s' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was…

Let the grid graph $G_{M\times N}$ denote the Cartesian product $K_M \square K_N$. For a fixed subgraph $H$ of a grid, we study the off-diagonal Ramsey number $\operatorname{gr}(H, K_k)$, which is the smallest $N$ such that any red/blue…