Ramsey theory refers to a large body of deep results in mathematics whose underlying philosophy is captured succinctly by the statement that â€œEvery large system contains a large well-organizedâ€¦ (More)

We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of mâ€¦ (More)

For a permutation Ï€, let Sn(Ï€) be the number of permutations on n letters avoiding Ï€. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that L(Ï€) = limnâ†’âˆž Sn(Ï€) 1/n exists and isâ€¦ (More)

Let G be a graph with n vertices and independence number Î±. Hadwigerâ€™s conjecture implies that G contains a clique minor of order at least n/Î±. In 1982, Duchet and Meyniel proved that this boundâ€¦ (More)

An intersection graph of curves in the plane is called a string graph. MatouÅ¡ek almost completely settled a conjecture of the authors by showing that every string graph with m edges admits a vertexâ€¦ (More)

For any sequence of positive integers j1 < j2 < Â· Â· Â· < jn, the k-tuples (ji, ji+1, ..., ji+kâˆ’1), i = 1, 2, . . . , nâˆ’k+1, are said to form a monotone path of length n. Given any integers n â‰¥ k â‰¥ 2â€¦ (More)

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 classicalâ€¦ (More)

Computing the maximum number of disjoint elements in a collection <i>C</i> of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment inâ€¦ (More)

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck/ logâˆ— k pairs of parts which are not Ç«-regular, where c, Ç«â€¦ (More)