Jacob Fox

Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements.

Overlap properties of geometric expanders

Abstract The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) ∈ (0, 1] such that no matter how we map the vertices of H into ℝd, there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. Motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, we address the question whether or not there exists a sequence of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree for which . Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any h, s and any ɛ > 0, there exists K = K(ɛ, h, s) satisfying the following condition. For any k ≧ K and for any semi-algebraic relation R on h-tuples of points in a Euclidean space ℝd with description complexity at most s, every finite set P ⫅ ℝd has a partition P = P1 ∪ ⋯ ∪ Pk into k parts of sizes as equal as possible such that all but at most an ɛ-fraction of the h-tuples (Pi1, … , Pih) have the property that either all h-tuples of points with one element in each Pij are related with respect to R or none of them are.

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.

Dependent random choice

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them.

THE NUMBER OF EDGES IN k-QUASI-PLANAR GRAPHS

A graph drawn in the plane is called

Coloring k k -free intersection graphs of geometric objects in the plane

k

A relative Szemerédi theorem

-quasi-planar if it does not contain

A minimum degree condition forcing complete graph immersion

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Extremal results in sparse pseudorandom graphs

pairwise crossing edges. It has been conjectured for a long time that for every fixed

A separator theorem for string graphs and its applications

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