arXiv subjects

Jacob Fox

Publications and source records attributed to Jacob Fox.

165 records · Page 10Linked to original sources

arXiv2007

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.

math.CO↗

arXiv2007

Ramsey numbers of sparse hypergraphs

We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree Δhas Ramsey number at most c(Δ, k)n, for an appropriate constant c(Δ, k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self-contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and Rödl. Moreover, our method demonstrates that, for k \geq 4, c(Δ, k) \leq 2^{2^{\Ddots^{2^{c Δ}}}}, where the tower is of height k and the constant c depends on k. It significantly improves on the Ackermann-type upper bound that arises from the regularity proofs, and we present a construction which shows that, at least in certain cases, this bound is not far from best possible. Our methods also allows us to prove quite sharp results on the Ramsey number of hypergraphs with at most m edges.

math.CO↗

arXiv2007

Ramsey-type problem for an almost monochromatic K_4

In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.

math.CO↗