Size Ramsey Number Research Articles

The connected size Ramsey number for matchings versus small disconnected graphs

Let F , G , and H be simple graphs. The notation F → ( G , H ) means that if all the edges of F are arbitrarily colored by red or blue, then there always exists either a red subgraph G or a blue subgraph H . The size Ramsey number of graph G and H , denoted by r ( G , H ) is the smallest integer k such that there is a graph F with k edges satisfying F → ( G , H ). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs F satisfying the above properties. This connected size Ramsey number of G and H is denoted by r c ( G , H ). We will derive an upper bound of r c ( n K 2 , H ), n ≥ 2 where H is 2 P m or 2 K 1, t , and find the exact values of r c ( n K 2 , H ), for some fixed n .

Ordered size Ramsey number of paths

An ordered graph is a simple graph with an ordering on its vertices. Define the ordered pathPn to be the monotone increasing path with n edges. The ordered size Ramsey numberr̃(Pr,Ps) is the minimum number m for which there exists an ordered graph H with m edges such that every two-coloring of the edges of H contains a red copy of Pr or a blue copy of Ps. For 2≤r≤s, we show 18r2s≤r̃(Pr,Ps)≤Cr2s(logs)3, where C>0 is an absolute constant. This problem is motivated by the recent results of Bucić et al. (2019) and Letzter and Sudakov (2019) for oriented graphs.

Size Ramsey numbers of stars versus cliques

AbstractThe size Ramsey number of two graphs and is the smallest integer such that there exists a graph on edges with the property that every red‐blue colouring of the edges of yields a red copy of or a blue copy of . In 1981, Erdős observed thaturn:x-wiley:03649024:media:jgt22453:jgt22453-math-0010 and he conjectured that this upper bound on is sharp. In 1983, Faudree and Sheehan extended this conjecture as follows:urn:x-wiley:03649024:media:jgt22453:jgt22453-math-0012 They proved the case . In 2001, Pikhurko showed that this conjecture is not true for and , by disproving the mentioned conjecture of Erdős. Here, we prove Faudree and Sheehan's conjecture for a given and .

On the restricted size Ramsey number involving a path P_3

On the restricted size Ramsey number involving a path P_3

Note on the Multicolour Size-Ramsey Number for Paths,

The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ 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 $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = \Omega(r^2 n)$.

The size Ramsey number of short subdivisions of bounded degree graphs

AbstractFor graphs G and F, write if any coloring of the edges of G with colors yields a monochromatic copy of the graph F. Suppose is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h times (that is, by replacing the edges of S by paths of length h + 1). We prove that there exists a graph G with no more than edges for which holds, provided that . We also extend this result to the case in which Q is a graph with maximum degree d on q vertices with the property that every pair of vertices of degree greater than 2 are distance at least h + 1 apart. This complements work of Pak regarding the size Ramsey number of “long subdivisions” of bounded degree graphs.

Size-Ramsey numbers of cycles versus a path

The size-Ramsey number Rˆ(F,H) of a family of graphs 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 coloring of the edges of G with two colors, say, red and blue, yields a red copy of a graph from F or a blue copy of H. In this paper we first focus on F=C≤cn, where C≤cn is the family of cycles of length at most cn, and H=Pn. In particular, we show that 2.00365n≤Rˆ(C≤n,Pn)≤31n. Using similar techniques, we also managed to analyze Rˆ(Cn,Pn), which was investigated before but until last year only by using the regularity method.

Monochromatic paths in random tournaments

AbstractWe prove that, with high probability, any 2‐edge‐coloring of a random tournament on n vertices contains a monochromatic path of length . This resolves a conjecture of Ben‐Eliezer, Krivelevich, and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.

Restricted Size Ramsey Number for 2K2 versus Dense Connected Graphs of Order Six

Let G and H be simple graphs. The Ramsey number r(G, H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H. The size Ramsey number for a pair of graphs G and H is the smallest number such that there exists a graph F with size satisfying the property that any red-blue coloring of the edges of F contains a red subgraph G or a blue subgraph H. Additionally, if the order of F in the size Ramsey number equals r(G, H), then it is called the restricted size Ramsey number. In 1972, Chvátal and Harary gave the Ramsey number for 2K2 versus any graph H with no isolates. In 1983, Harary and Miller started the investigation of the (restricted) size Ramsey number for some pairs of small graphs with order at most four. In 1983, Faudree and Sheehan continued the investigation and summarized the complete results on the (restricted) size Ramsey number for all pairs of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order at most five. Lately, we investigate the restricted size Ramsey number for 2K2 versus all connected graphs of order five. In this work, we continue the investigation on the restricted size Ramsey number for a pair of small graphs. In particularly, for 2K2 versus dense connected graph of order six.

Publisher Correction to: Connected size Ramsey number for matchings vs. small stars or cycles

Publisher Correction to: Connected size Ramsey number for matchings vs. small stars or cycles

On a Question of Erdös and Faudree on the Size Ramsey Numbers

For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ with $m$ edges such that in any edge coloring of $G$ with two colors red and blue, there is either a red copy of $G_1$ or a blue copy of $G_2$. In 1981, Erdos and Faudree investigated the size Ramsey number $\hat{R}(K_n,tK_2)$, where $K_n$ is a complete graph on $n$ vertices and $tK_2$ is a matching of size $t$. They obtained the value of $\hat{R}(K_n,tK_2)$ when $n\geq 4t-1$ as well as for $t=2$ and asked for the behavior of these numbers when $t$ is much larger than $n$. In this regard, they posed the following interesting question: For every positive integer $n$, is it true that $\lim_{t\to \infty} ({\hat{R}(K_n,tK_2)}/{t\, \hat{R}(K_n,K_2)})= \min\{\binom{n+2t-2}{2}/ {t\binom{n}{2}}\mid t\in \mathbb{N}\}?$ In this paper, we obtain the exact value of $\hat{R}(K_n,tK_2)$ for every pair of positive integers $ n,t$, and as a byproduct, we give an affirmati...

On the Size-Ramsey Number of Tight Paths

For any $r\geq 2$ and $k\geq 3$, the $r$-color size-Ramsey number $\hat R(\mathcal{G},r)$ of a $k$-uniform hypergraph $\mathcal{G}$ is the smallest integer $m$ such that there exists a $k$-uniform ...

Connected size Ramsey number for matchings vs. small stars or cycles

The notation $$F\rightarrow (G,H)$$ means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The connected size Ramsey number $$\hat{r}_c(G,H)$$ of graphs G and H is the minimum size of a connected graph F satisfying $$F\rightarrow (G,H)$$ . For $$m \ge 2,$$ the graph consisting of m independent edges is called a matching and is denoted by $$mK_2$$ . In 1981, Erdos and Faudree determined the size Ramsey numbers for the pair $$(mK_2, K_{1,t})$$ . They showed that the disconnected graph $$mK_{1,t} \rightarrow (mK_2,K_{1,t})$$ for $$ t,m \ge 1$$ . In this paper, we will determine the connected size Ramsey number $$\hat{r}_c(nK_2, K_{1,3})$$ for $$n\ge 2$$ and $$\hat{r}_c(3K_2, C_4)$$ . We also derive an upper bound of the connected size Ramsey number $$\hat{r}_c(nK_2, C_4),$$ for $$n\ge 4$$ .

A class of Ramsey-extremal hypergraphs

In 1991, McKay and Radziszowski proved that, however each 3-subset of a 13-set is assigned one of two colours, there is some 4-subset whose four 3-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey number known for hypergraphs. In this article, we find all the extremal colourings of the 3-subsets of a 12-set and list some of their properties. Using the catalogue, we provide an answer to a question of Dudek, Fleur, Mubayi and Rodl about the size-Ramsey numbers of hypergraphs.

A strict upper bound for size multipartite Ramsey numbers of paths versus stars

<p>Let $P_n$ represent the path of size $n$. Let $K_{1,m-1}$ represent a star of size $m$ and be denoted by $S_{m}$. Given a two coloring of the edges of a complete graph $K_{j \times s}$ we say that $K_{j \times s}\rightarrow (P_n,S_{m+1})$ if there is a copy of $P_n$ in the first color or a copy of $S_{m+1}$ in the second color. The size Ramsey multipartite number $m_j(P_n, S_{m+1})$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (P_n,S_{m+1})$. Given $j,n,m$ if $s=\left\lceil \dfrac{n+m-1-k}{j-1} \right\rceil$, in this paper, we show that the size Ramsey numbers $m_j(P_n,S_{m+1})$ is bounded above by $s$ for $k=\left\lceil \dfrac{n-1}{j} \right\rceil$. Given $j\ge 3$ and $s$, we will obtain an infinite class $(n,m)$ that achieves this upper bound $s$. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.</p>

Monochromatic paths in random tournaments

We prove that, with high probability, any 2-edge colouring of a random tournament on n vertices contains a monochromatic path of length Ω(n/log⁡n). This resolves a conjecture of Ben-Eliezer, Krivelevich and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.

Restricted size Ramsey number for path of order three versus graph of order five

Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for a pair of graph $G$ and $H$ is the smallest number $\hat{r}$ such that there exists a graph $F$ with size $\hat{r}$ satisfying the property that any red-blue coloring of edges of $F$ contains a red subgraph $G$ or a blue subgraph $H$. Additionally, if the order of $F$ in the size Ramsey number is $r(G,H)$, then it is called the restricted size Ramsey number. In 1983, Harary and Miller started to find the (restricted) size Ramsey number for any pair of small graphs with order at most four. Faudree and Sheehan (1983) continued Harary and Miller's works and summarized the complete results on the (restricted) size Ramsey number for any pair of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for any pair of small forests with order at most five. To continue their works, we investigate the restricted size Ramsey number for a path of order three versus connected graph of order five.

On some Multicolor Ramsey Properties of Random Graphs

The size-Ramsey number ${R}{F}$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any coloring of the edges of $G$ with two colors yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le {R}(){P_n} \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This improves the previous lower bound, ${R}{P_n} \ge (1+\sqrt{2})n-O(1)$, due to Bollobas, and the upper bound, ${R}(){P_n} \le 91n$, due to Letzter. Next we study long monochromatic paths in an edge-colored random graph $\mathcal{G}(n,p)$ with $pn \to \infty$. Let $\alpha > 0$ be an arbitrarily small constant. Recently, Letzter showed that asymptotically almost surely (a.a.s.) any $2$-edge coloring of $\mathcal{G}(n,p)$ yields a monochromatic path of length $(2/3-\alpha)n$, which is optimal. Extending this result, we show that ...

Size multipartite Ramsey numbers for small paths versus books

Given $j \ge 2$, for graphs $G$ and $H$, the size Ramsey multipartite number $m_j(G, H)$ is defined as the smallest natural number $t$ such that any blue red coloring of the edges of the graph $K_{j \times t}$, necessarily containes a red $G$ or a blue $H$ as subgraphs. Let the book with $n$ pages is defined as the graph $K_1 + K_{1,n}$ and denoted by $B_n$. In this paper, we obtain the exact values of the size Ramsey numbers $m_j(P_3, H)$ for $j \ge 3$ where $H$ is a book $B_n$. We also derive some upper and lower bounds for the size Ramsey numbers $m_j(P_4, H)$ where $H$ is a book $B_n$.

Size multipartite Ramsey numbers for stripes versus small cycles

For simple graphs $G_1$ and $G_2$, the size Ramsey multipartite number $m_j(G_1, G_2)$ is defined as the smallest natural number $s$ such that any arbitrary two coloring of the graph $K_{j \times s}$ using the colors red and blue, contains a red $G_1$ or a blue $G_2$ as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers $m_j(nK_2, C_m)$ for $j \ge 2$ and $m \in \{3,4,5,6\}$.