On-line Ramsey numbers

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey number \tilde{r}(G) of G. Our main result, relating to the conjecture that \tilde{r}(K_t) = o(\binom{r(t)}{2}), is that there exists a constant c > 1 such that \tilde{r}(K_t) \leq c^{-t} \binom{r(t)}{2} for infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a constant c such that \tilde{r}(K_t) \leq t^{-c \frac{\log t}{\log \log t}} 4^t. Finally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph K_{t,t}.Comment: 11 page

On-line Ramsey numbers of paths and cycles

Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey number $\tilde{r}(G,H)$. In this paper, we consider the case where $G$ is a path $P_k$. We prove that $\tilde{r}(P_3, P_{\ell+1}) = \lceil 5\ell/4 \rceil = \tilde{r}(P_3, C_\ell)$ for all $\ell \ge 5$, and determine $\tilde{r}(P_4, P_{\ell+1}$) up to an additive constant for all $\ell \ge 3$. We also prove some general lower bounds for on-line Ramsey numbers of the form $\tilde{r}(P_{k+1},H)$.Comment: Preprin

Proof of a conjecture on induced subgraphs of Ramsey graphs

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and S\'{o}s conjectured that in any C-Ramsey graph there are $\Omega\left(n^{5/2}\right)$ induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture

On-line Ramsey numbers for paths and stars

Graphs and Algorithm

Hypergraph Ramsey numbers

The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3 and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it gives the first superexponential lower bound for r_3(s,n), answering an open question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of the triples of an N-element set contains a monochromatic set of size n. Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \geq 2^{n^{c \log n}}. Finally, we make some progress on related hypergraph Ramsey-type problems

On the Geometric Ramsey Number of Outerplanar Graphs

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $2n$ vertices are bounded by $O(n^{3})$ and $O(n^{10})$, in the convex and general case, respectively. We then apply similar methods to prove an $n^{O(\log(n))}$ upper bound on the Ramsey number of a path with $n$ ordered vertices.Comment: 15 pages, 7 figure

On the minimum degree of minimal Ramsey graphs for multiple colours

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) = r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r