On the size-Ramsey number of cycles

For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $H$ contains a monochromatic copy of $G_i$ of color $i$ for some $1\leq i\leq k$. We denote $\hat{R}(G_1,\ldots,G_k)$ by $\hat{R}_{k}(G)$ when $G_1=\cdots=G_k=G$. Haxell, Kohayakawa and \L{}uczak showed that the size-Ramsey number of a cycle $C_n$ is linear in $n$ i.e. $\hat{R}_{k}(C_{n})\leq c_k n$ for some constant $c_k$. Their proof, however, is based on the regularity lemma of Szemer\'{e}di and so no specific constant $c_k$ is known. In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of $\hat{R}_{k}(C_{n})\leq c_k n$, avoiding the use of the regularity lemma, where $c_k$ is exponential and doubly-exponential in $k$, when $n$ is even and odd, respectively. In particular, we show that for sufficiently large $n$ we have $\hat{R}(C_{n},C_{n}) \leq 10^5\times cn,$ where $c=6.5$ if $n$ is even and $c=1989$ otherwise

Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric

Bipartite, Size, and Online Ramsey Numbers of Some Cycles and Paths

The basic premise of Ramsey Theory states that in a sufficiently large system, complete disorder is impossible. One instance from the world of graph theory says that given two fixed graphs F and H, there exists a finitely large graph G such that any red/blue edge coloring of the edges of G will produce a red copy of F or a blue copy of H. Much research has been conducted in recent decades on quantifying exactly how large G must be if we consider different classes of graphs for F and H. In this thesis, we explore several Ramsey- type problems with a particular focus on paths and cycles. We first examine the bipartite size Ramsey number of a path on n vertices, bˆr(Pn), and give an upper bound using a random graph construction motivated by prior upper bound improvements in similar problems. Next, we consider the size Ramsey number Rˆ (C, Pn) and provide a significant improvement to the upper bound using a very structured graph, the cube of a path, as opposed to a random construction. We also prove a small improvement to the lower bound and show that the r-colored version of this problem is asymptotically linear in rn. Lastly, we give an upper bound for the online Ramsey number R˜ (C, Pn)

Online size Ramsey numbers: Odd cycles vs connected graphs

Given two graph families $\mathcal H_1$ and $\mathcal H_2$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder is trying to force Painter to create as soon as possible a red copy of a graph from $\mathcal H_1$ or a blue copy of a graph from $\mathcal H_2$. The online (size) Ramsey number $\tilde{r}(\mathcal H_1,\mathcal H_2)$ is the smallest number of rounds in the game provided Builder and Painter play optimally. We prove that if $\mathcal H_1$ is the family of all odd cycles and $\mathcal H_2$ is the family of all connected graphs on $n$ vertices and $m$ edges, then $\tilde{r}(\mathcal H_1,\mathcal H_2)\ge \varphi n + m-2\varphi+1$, where $\varphi$ is the golden ratio, and for $n\ge 3$, $m\le (n-1)^2/4$ we have $\tilde{r}(\mathcal H_1,\mathcal H_2)\le n+2m+O(\sqrt{m-n+1})$. We also show that $\tilde{r}(C_3,P_n)\le 3n-4$ for $n\ge 3$. As a consequence we get $2.6n-3\le \tilde{r}(C_3,P_n)\le 3n-4$ for every $n\ge 3$.Comment: 14 pages, 0 figures; added appendix containing intuition behind the potential function used for lower bound; corrected typos and added a few clarification

Packings and coverings with Hamilton cycles and on-line Ramsey theory

A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area. • Let ε > 0, and let $G$ be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of $G$ with edge-disjoint Hamilton cycles. • Let $T$ be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of $T$ with edge-disjoint Hamilton cycles. • Let log $^1$$^1$$^7$ $n$/$n$≤$p$≤1-$n$$^-$$^1$$^/$$^8$. We prove that $G$$_n$$_,$$_p$ may a.a.s be covered by a set of ⌈Δ($G$$_n$$_,$$_p$)/2⌉ Hamilton cycles, which is clearly best possible. In addition, we consider some problems in on-line Ramsey theory. Let r($G$,$H$) denote the on-line Ramsey number of $G$ and $H$. We conjecture the exact values of r ($P$$_k$,$P$$_ℓ$) for all $k$≤ℓ. We prove this conjecture for $k$=2, prove it to within an additive error of 10 for $k$=3, and prove an asymptotically tight lower bound for $k$=4. We also determine r($P$$_3$,$C$$_ℓ$ exactly for all ℓ