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...

Let P1, P2, . . . , Pd+1 be pairwise disjoint n-element point sets in general position in d-space. It is shown that there exist a point O and suitable subsets Qi ⊆ Pi (i = 1, 2, . . . , d + 1) such that |Qi| ≥ cd|Pi|, and every d-dimensional simplex with exactly one vertex in each Qi contains O in its interior. Here cd is a positive constant depending only on d.

We consider anti-Ramsey type results. For a given coloring ∆ of the k-element subsets of an n-element set X, where two k-element sets with nonempty intersection are colored differently, let inj∆(k, n) be the largest size of a subset Y ⊆ X, such that the k-element subsets of Y are colored pairwise differently. Taking the minimum over all colorings, i.e. inj(k, n) = min∆ {inj∆(k, n)}, it is shown...

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...