A Ramsey type result for oriented trees

A Ramsey type result for oriented trees

Given positive integers h and k, denote by r(h, k) the smallest integer n such that in any k-coloring of the edges of a tournament on more than n vertices there is a monochromatic copy of every oriented tree on h vertices. We prove that r(h, k) = (h− 1) for all k sufficiently large (k = Θ(h log h) suffices). The bound (h − 1) is tight. The related parameter r∗(h, k) where some color contains al...

Ramsey-type results for oriented trees

A Ramsey-type Result for Convex Sets

Given a family of n convex compact sets in the plane, one can always choose n of them which are either pairwise disjoint or pairwise intersecting. On the other hand, there exists a family of n segments in the plane such that the maximum size of a subfamily with pairwise disjoint or pairwise intersecting elements in n*"* ^ n-.

A Ramsey-type result for the hypercube

We prove that for every fixed k and ` ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2`. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, i.e., have the property that for every k, any k-edge coloring of a sufficiently large...

A Ramsey-Type Result for Geometric ℓ-Hypergraphs

Let n ≥ l ≥ 2 and q ≥ 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hy...