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

Combinatorial theorems relative to a random set

We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

The Ramsey number of dense graphs

The Ramsey number r(H) of a graph H is the smallest number n such that, in any two-colouring of the edges of K_n, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t vertices and density \r, proving that r(H) \leq 2^{c \sqrt{\r} \log (2/\r) t}. We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree \r t and the Ramsey number of random graphs in \mathcal{G}(t, \r), that is, graphs on t vertices where each edge has been chosen independently with probability \r.Comment: 15 page

Finite reflection groups and graph norms

Given a graph $H$ on vertex set $\{1,2,\cdots, n\}$ and a function $f:[0,1]^2 \rightarrow \mathbb{R}$, define \begin{align*} \|f\|_{H}:=\left\vert\int \prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*} where $\mu$ is the Lebesgue measure on $[0,1]$. We say that $H$ is norming if $\|\cdot\|_H$ is a semi-norm. A similar notion $\|\cdot\|_{r(H)}$ is defined by $\|f\|_{r(H)}:=\||f|\|_{H}$ and $H$ is said to be weakly norming if $\|\cdot\|_{r(H)}$ is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorphisms is weakly norming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture.Comment: 29 page

Lines in Euclidean Ramsey theory

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $K$.Comment: 7 page

A sequence of triangle-free pseudorandom graphs

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.Comment: 6 page

An extremal theorem in the hypercube

The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_n, H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.Comment: 6 page

Large subgraphs without complete bipartite graphs

In this note, we answer the following question of Foucaud, Krivelevich and Perarnau. What is the size of the largest $K_{r,s}$-free subgraph one can guarantee in every graph $G$ with $m$ edges? We also discuss the analogous problem for hypergraphs.Comment: 4 page