相关论文: Asymmetric Ramsey Properties of Random Graphs for …

For fixed graphs $F_1,\dotsc,F_r$, we prove an upper bound on the threshold function for the property that $G(n,p) \to (F_1,\dotsc,F_r)$. This establishes the $1$-statement of a conjecture of Kohayakawa and Kreuter.

In this paper, we study asymmetric Ramsey properties of the random graph $G_{n,p}$. Let $r \in \mathbb{N}$ and $H_1, \ldots, H_r$ be graphs. We write $G_{n,p} \to (H_1, \ldots, H_r)$ to denote the property that whenever we colour the edges…

Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\dots,H_r)$ of graphs if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i \in [r]$. A famous conjecture of Kohayakawa and…

A graph $G$ is said to be Ramsey for a tuple of graphs $(H_1,\dots,H_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i$. A fundamental question at the intersection of Ramsey…

For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and…

A graph $G$ is $(a,b)$-sparse if every nonempty subgraph $H$ satisfies $e(H) \leq a v(H) - b$. We are interested in the conditions under which an $(a,b)$-sparse graph can be partitioned $E(G) = E(G_1) \cup E(G_2)$ such that for $i \in…

A well-known result of R\"odl and Ruci\'nski states that for any graph $H$ there exists a constant $C$ such that if $p \geq C n^{- 1/m_2(H)}$, then the random graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of its edges…

For graphs $F$ and $G$, let $F\to G$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Denote by ${\cal G}(N,p)$ the random graph space of order $N$ and edge probability $p$. Using the regularity method, one can…

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, \v{S}kori\'{c} and…

Given graphs $F,H$ and $G$, we say that $G$ is $(F,H)_v$-Ramsey if every red/blue vertex colouring of $G$ contains a red copy of $F$ or a blue copy of $H$. Results of {\L}uczak, Ruci\'nski and Voigt and, subsequently, Kreuter determine the…

Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erd\H{o}s, and Spencer showed that…

We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In this paper we study the threshold for the…

For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for…

For graphs $G$ and $H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H $ if any proper edge-coloring of $G$ contains a rainbow copy of $H$, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last…

Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey…

Let $H_1$ and $H_2$ be graphs. A graph $G$ has the constrained Ramsey property for $(H_1,H_2)$ if every edge-colouring of $G$ contains either a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. Our main result gives a 0-statement for…

Let $R(C_n)$ be the Ramsey number of the cycle on $n$ vertices. We prove that, for some $C > 0$, with high probability every $2$-colouring of the edges of $G(N,p)$ has a monochromatic copy of $C_n$, as long as $N\geq R(C_n) + C/p$ and $p…

A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq…

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…

In this note we consider a Ramsey property of random $d$-regular graphs, $\mathcal{G}(n,d)$. Let $r\ge 2$ be fixed. Then w.h.p. the edges of $\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the…