David Conlon

Hypergraph Ramsey numbers

The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N -element set contains 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 rk(s, n) for k ≥ 3 and s fixed. In particular, we show that r3(s, n) ≤ 2 s−2 log , which improves by a factor of ns−2/polylogn the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant c > 0 such that r3(s, n) ≥ 2 sn log( n s +1) for all 4 ≤ s ≤ n. For constant s, it gives the first superexponential lower bound for r3(s, n), answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the 3-color Ramsey number r3(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 Erdős and Hajnal, we show that r3(n, n, n) ≥ 2 c log n . Finally, we make some progress on related hypergraph Ramsey-type problems.

WEAK QUASI-RANDOMNESS FOR UNIFORM HYPERGRAPHS

We study quasi-random properties of k-uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will nd several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs. Moreover, let Kk be the complete graph on k vertices and M(k) the line graph of the graph of the k-dimensional hypercube. We will show that the pair of graphs (Kk;M(k)) has the property that if the number of copies of both Kk and M(k) in another graph G are as expected in the random graph of density d, then G is quasi-random (in the sense of the Chung-Graham-Wilson theorem) with density close to d.

A relative Szemerédi theorem

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for k-term arithmetic progressions in pseudorandom subsets of

Extremal results in sparse pseudorandom graphs

{\mathbb{Z}_N}

Hypergraph packing and sparse bipartite ramsey numbers

ZN of density

On-line Ramsey Numbers

{N^{-c_k}}